Theorem propeqop 4970

Description: Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)

Hypotheses

Ref	Expression
snopeqop.a	|- A e. _V
snopeqop.b	|- B e. _V
snopeqop.c	|- C e. _V
snopeqop.d	|- D e. _V
propeqop.e	|- E e. _V
propeqop.f	|- F e. _V

Assertion

Ref	Expression
propeqop	|- ( { <. A , B >. , <. C , D >. } = <. E , F >. <-> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )

Proof of Theorem propeqop

Step	Hyp	Ref	Expression
1		snopeqop.a	. . . . 5 |- A e. _V
2		snopeqop.b	. . . . 5 |- B e. _V
3		propeqop.e	. . . . 5 |- E e. _V
4	1, 2, 3	opeqsn 4967	. . . 4 |- ( <. A , B >. = { E } <-> ( A = B /\ E = { A } ) )
5		snopeqop.c	. . . . 5 |- C e. _V
6		snopeqop.d	. . . . 5 |- D e. _V
7		propeqop.f	. . . . 5 |- F e. _V
8	5, 6, 3, 7	opeqpr 4968	. . . 4 |- ( <. C , D >. = { E , F } <-> ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) )
9	4, 8	anbi12i 733	. . 3 |- ( ( <. A , B >. = { E } /\ <. C , D >. = { E , F } ) <-> ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) )
10	1, 2, 3, 7	opeqpr 4968	. . . 4 |- ( <. A , B >. = { E , F } <-> ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) )
11	5, 6, 3	opeqsn 4967	. . . 4 |- ( <. C , D >. = { E } <-> ( C = D /\ E = { C } ) )
12	10, 11	anbi12i 733	. . 3 |- ( ( <. A , B >. = { E , F } /\ <. C , D >. = { E } ) <-> ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) )
13	9, 12	orbi12i 543	. 2 |- ( ( ( <. A , B >. = { E } /\ <. C , D >. = { E , F } ) \/ ( <. A , B >. = { E , F } /\ <. C , D >. = { E } ) ) <-> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) )
14		eqcom 2629	. . 3 |- ( { <. A , B >. , <. C , D >. } = <. E , F >. <-> <. E , F >. = { <. A , B >. , <. C , D >. } )
15		opex 4932	. . . 4 |- <. A , B >. e. _V
16		opex 4932	. . . 4 |- <. C , D >. e. _V
17	3, 7, 15, 16	opeqpr 4968	. . 3 |- ( <. E , F >. = { <. A , B >. , <. C , D >. } <-> ( ( <. A , B >. = { E } /\ <. C , D >. = { E , F } ) \/ ( <. A , B >. = { E , F } /\ <. C , D >. = { E } ) ) )
18	14, 17	bitri 264	. 2 |- ( { <. A , B >. , <. C , D >. } = <. E , F >. <-> ( ( <. A , B >. = { E } /\ <. C , D >. = { E , F } ) \/ ( <. A , B >. = { E , F } /\ <. C , D >. = { E } ) ) )
19		simpl 473	. . . . . . . . 9 |- ( ( A = B /\ F = { A , D } ) -> A = B )
20		simpr 477	. . . . . . . . 9 |- ( ( A = C /\ E = { A } ) -> E = { A } )
21	19, 20	anim12i 590	. . . . . . . 8 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> ( A = B /\ E = { A } ) )
22		sneq 4187	. . . . . . . . . . . . 13 |- ( A = C -> { A } = { C } )
23	22	eqeq2d 2632	. . . . . . . . . . . 12 |- ( A = C -> ( E = { A } <-> E = { C } ) )
24	23	biimpa 501	. . . . . . . . . . 11 |- ( ( A = C /\ E = { A } ) -> E = { C } )
25	24	adantl 482	. . . . . . . . . 10 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> E = { C } )
26		preq1 4268	. . . . . . . . . . . . . . 15 |- ( A = C -> { A , D } = { C , D } )
27	26	adantr 481	. . . . . . . . . . . . . 14 |- ( ( A = C /\ E = { A } ) -> { A , D } = { C , D } )
28	27	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( ( A = C /\ E = { A } ) -> ( F = { A , D } <-> F = { C , D } ) )
29	28	biimpcd 239	. . . . . . . . . . . 12 |- ( F = { A , D } -> ( ( A = C /\ E = { A } ) -> F = { C , D } ) )
30	29	adantl 482	. . . . . . . . . . 11 |- ( ( A = B /\ F = { A , D } ) -> ( ( A = C /\ E = { A } ) -> F = { C , D } ) )
31	30	imp 445	. . . . . . . . . 10 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> F = { C , D } )
32	25, 31	jca 554	. . . . . . . . 9 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> ( E = { C } /\ F = { C , D } ) )
33	32	orcd 407	. . . . . . . 8 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) )
34	21, 33	jca 554	. . . . . . 7 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) )
35	34	orcd 407	. . . . . 6 |- ( ( ( A = B /\ F = { A , D } ) /\ ( A = C /\ E = { A } ) ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) )
36	35	ex 450	. . . . 5 |- ( ( A = B /\ F = { A , D } ) -> ( ( A = C /\ E = { A } ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) ) )
37		simpr 477	. . . . . . . . . . 11 |- ( ( A = D /\ F = { A , B } ) -> F = { A , B } )
38	20, 37	anim12i 590	. . . . . . . . . 10 |- ( ( ( A = C /\ E = { A } ) /\ ( A = D /\ F = { A , B } ) ) -> ( E = { A } /\ F = { A , B } ) )
39	38	ancoms 469	. . . . . . . . 9 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> ( E = { A } /\ F = { A , B } ) )
40	39	orcd 407	. . . . . . . 8 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) )
41		simpl 473	. . . . . . . . . . . . 13 |- ( ( A = C /\ E = { A } ) -> A = C )
42	41	eqeq1d 2624	. . . . . . . . . . . 12 |- ( ( A = C /\ E = { A } ) -> ( A = D <-> C = D ) )
43	42	biimpcd 239	. . . . . . . . . . 11 |- ( A = D -> ( ( A = C /\ E = { A } ) -> C = D ) )
44	43	adantr 481	. . . . . . . . . 10 |- ( ( A = D /\ F = { A , B } ) -> ( ( A = C /\ E = { A } ) -> C = D ) )
45	44	imp 445	. . . . . . . . 9 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> C = D )
46	23	biimpd 219	. . . . . . . . . . . 12 |- ( A = C -> ( E = { A } -> E = { C } ) )
47	46	a1dd 50	. . . . . . . . . . 11 |- ( A = C -> ( E = { A } -> ( ( A = D /\ F = { A , B } ) -> E = { C } ) ) )
48	47	imp 445	. . . . . . . . . 10 |- ( ( A = C /\ E = { A } ) -> ( ( A = D /\ F = { A , B } ) -> E = { C } ) )
49	48	impcom 446	. . . . . . . . 9 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> E = { C } )
50	45, 49	jca 554	. . . . . . . 8 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> ( C = D /\ E = { C } ) )
51	40, 50	jca 554	. . . . . . 7 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) )
52	51	olcd 408	. . . . . 6 |- ( ( ( A = D /\ F = { A , B } ) /\ ( A = C /\ E = { A } ) ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) )
53	52	ex 450	. . . . 5 |- ( ( A = D /\ F = { A , B } ) -> ( ( A = C /\ E = { A } ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) ) )
54	36, 53	jaoi 394	. . . 4 |- ( ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) -> ( ( A = C /\ E = { A } ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) ) )
55	54	impcom 446	. . 3 |- ( ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) -> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) )
56		eqeq1 2626	. . . . . . . . . . . . 13 |- ( E = { C } -> ( E = { A } <-> { C } = { A } ) )
57	5	sneqr 4371	. . . . . . . . . . . . . 14 |- ( { C } = { A } -> C = A )
58	57	eqcomd 2628	. . . . . . . . . . . . 13 |- ( { C } = { A } -> A = C )
59	56, 58	syl6bi 243	. . . . . . . . . . . 12 |- ( E = { C } -> ( E = { A } -> A = C ) )
60	59	adantr 481	. . . . . . . . . . 11 |- ( ( E = { C } /\ F = { C , D } ) -> ( E = { A } -> A = C ) )
61		eqeq1 2626	. . . . . . . . . . . . 13 |- ( E = { C , D } -> ( E = { A } <-> { C , D } = { A } ) )
62	5, 6, 1	preqsn 4393	. . . . . . . . . . . . . 14 |- ( { C , D } = { A } <-> ( C = D /\ D = A ) )
63		eqtr 2641	. . . . . . . . . . . . . . 15 |- ( ( C = D /\ D = A ) -> C = A )
64	63	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( C = D /\ D = A ) -> A = C )
65	62, 64	sylbi 207	. . . . . . . . . . . . 13 |- ( { C , D } = { A } -> A = C )
66	61, 65	syl6bi 243	. . . . . . . . . . . 12 |- ( E = { C , D } -> ( E = { A } -> A = C ) )
67	66	adantr 481	. . . . . . . . . . 11 |- ( ( E = { C , D } /\ F = { C } ) -> ( E = { A } -> A = C ) )
68	60, 67	jaoi 394	. . . . . . . . . 10 |- ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) -> ( E = { A } -> A = C ) )
69	68	com12 32	. . . . . . . . 9 |- ( E = { A } -> ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) -> A = C ) )
70	69	adantl 482	. . . . . . . 8 |- ( ( A = B /\ E = { A } ) -> ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) -> A = C ) )
71	70	impcom 446	. . . . . . 7 |- ( ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) /\ ( A = B /\ E = { A } ) ) -> A = C )
72		simpr 477	. . . . . . . 8 |- ( ( A = B /\ E = { A } ) -> E = { A } )
73	72	adantl 482	. . . . . . 7 |- ( ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) /\ ( A = B /\ E = { A } ) ) -> E = { A } )
74	71, 73	jca 554	. . . . . 6 |- ( ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) /\ ( A = B /\ E = { A } ) ) -> ( A = C /\ E = { A } ) )
75		simpl 473	. . . . . . . . . . 11 |- ( ( A = B /\ E = { A } ) -> A = B )
76	75	adantr 481	. . . . . . . . . 10 |- ( ( ( A = B /\ E = { A } ) /\ ( E = { C } /\ F = { C , D } ) ) -> A = B )
77		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 |- ( E = { A } -> ( E = { C } <-> { A } = { C } ) )
78	1	sneqr 4371	. . . . . . . . . . . . . . . . . . 19 |- ( { A } = { C } -> A = C )
79	78	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( { A } = { C } -> C = A )
80	77, 79	syl6bi 243	. . . . . . . . . . . . . . . . 17 |- ( E = { A } -> ( E = { C } -> C = A ) )
81	80	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( A = B /\ E = { A } ) -> ( E = { C } -> C = A ) )
82	81	impcom 446	. . . . . . . . . . . . . . 15 |- ( ( E = { C } /\ ( A = B /\ E = { A } ) ) -> C = A )
83	82	preq1d 4274	. . . . . . . . . . . . . 14 |- ( ( E = { C } /\ ( A = B /\ E = { A } ) ) -> { C , D } = { A , D } )
84	83	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( ( E = { C } /\ ( A = B /\ E = { A } ) ) -> ( F = { C , D } <-> F = { A , D } ) )
85	84	biimpd 219	. . . . . . . . . . . 12 |- ( ( E = { C } /\ ( A = B /\ E = { A } ) ) -> ( F = { C , D } -> F = { A , D } ) )
86	85	impancom 456	. . . . . . . . . . 11 |- ( ( E = { C } /\ F = { C , D } ) -> ( ( A = B /\ E = { A } ) -> F = { A , D } ) )
87	86	impcom 446	. . . . . . . . . 10 |- ( ( ( A = B /\ E = { A } ) /\ ( E = { C } /\ F = { C , D } ) ) -> F = { A , D } )
88	76, 87	jca 554	. . . . . . . . 9 |- ( ( ( A = B /\ E = { A } ) /\ ( E = { C } /\ F = { C , D } ) ) -> ( A = B /\ F = { A , D } ) )
89	88	ex 450	. . . . . . . 8 |- ( ( A = B /\ E = { A } ) -> ( ( E = { C } /\ F = { C , D } ) -> ( A = B /\ F = { A , D } ) ) )
90		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( D = A <-> A = D )
91	90	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 |- ( D = A -> A = D )
92	91	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( C = D /\ D = A ) -> A = D )
93	92	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> A = D )
94	93	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( C = D /\ D = A ) /\ A = B ) /\ F = { C } ) -> A = D )
95		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> A = B )
96	64	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( C = D /\ D = A ) -> ( A = B <-> C = B ) )
97	96	biimpa 501	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> C = B )
98	97	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> B = C )
99	1, 2, 5	preqsn 4393	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )
100	95, 98, 99	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> { A , B } = { C } )
101	100	eqcomd 2628	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> { C } = { A , B } )
102	101	eqeq2d 2632	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> ( F = { C } <-> F = { A , B } ) )
103	102	biimpa 501	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( C = D /\ D = A ) /\ A = B ) /\ F = { C } ) -> F = { A , B } )
104	94, 103	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( C = D /\ D = A ) /\ A = B ) /\ F = { C } ) -> ( A = D /\ F = { A , B } ) )
105	104	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( ( C = D /\ D = A ) /\ A = B ) -> ( F = { C } -> ( A = D /\ F = { A , B } ) ) )
106	105	ex 450	. . . . . . . . . . . . . . 15 |- ( ( C = D /\ D = A ) -> ( A = B -> ( F = { C } -> ( A = D /\ F = { A , B } ) ) ) )
107	106	com23 86	. . . . . . . . . . . . . 14 |- ( ( C = D /\ D = A ) -> ( F = { C } -> ( A = B -> ( A = D /\ F = { A , B } ) ) ) )
108	62, 107	sylbi 207	. . . . . . . . . . . . 13 |- ( { C , D } = { A } -> ( F = { C } -> ( A = B -> ( A = D /\ F = { A , B } ) ) ) )
109	61, 108	syl6bi 243	. . . . . . . . . . . 12 |- ( E = { C , D } -> ( E = { A } -> ( F = { C } -> ( A = B -> ( A = D /\ F = { A , B } ) ) ) ) )
110	109	com23 86	. . . . . . . . . . 11 |- ( E = { C , D } -> ( F = { C } -> ( E = { A } -> ( A = B -> ( A = D /\ F = { A , B } ) ) ) ) )
111	110	imp 445	. . . . . . . . . 10 |- ( ( E = { C , D } /\ F = { C } ) -> ( E = { A } -> ( A = B -> ( A = D /\ F = { A , B } ) ) ) )
112	111	com13 88	. . . . . . . . 9 |- ( A = B -> ( E = { A } -> ( ( E = { C , D } /\ F = { C } ) -> ( A = D /\ F = { A , B } ) ) ) )
113	112	imp 445	. . . . . . . 8 |- ( ( A = B /\ E = { A } ) -> ( ( E = { C , D } /\ F = { C } ) -> ( A = D /\ F = { A , B } ) ) )
114	89, 113	orim12d 883	. . . . . . 7 |- ( ( A = B /\ E = { A } ) -> ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) -> ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
115	114	impcom 446	. . . . . 6 |- ( ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) /\ ( A = B /\ E = { A } ) ) -> ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) )
116	74, 115	jca 554	. . . . 5 |- ( ( ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) /\ ( A = B /\ E = { A } ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
117	116	ancoms 469	. . . 4 |- ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
118		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( E = { C } -> ( E = { A , B } <-> { C } = { A , B } ) )
119		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( { C } = { A , B } <-> { A , B } = { C } )
120	119, 99	bitri 264	. . . . . . . . . . . . . . . . 17 |- ( { C } = { A , B } <-> ( A = B /\ B = C ) )
121		eqtr 2641	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A = B /\ B = C ) -> A = C )
122	121	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A = B /\ B = C ) /\ E = { C } ) -> A = C )
123	121	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( A = B /\ B = C ) -> C = A )
124		sneq 4187	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( C = A -> { C } = { A } )
125	123, 124	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A = B /\ B = C ) -> { C } = { A } )
126	125	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A = B /\ B = C ) -> ( E = { C } <-> E = { A } ) )
127	126	biimpa 501	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A = B /\ B = C ) /\ E = { C } ) -> E = { A } )
128	122, 127	jca 554	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( A = B /\ B = C ) /\ E = { C } ) -> ( A = C /\ E = { A } ) )
129	128	ex 450	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A = B /\ B = C ) -> ( E = { C } -> ( A = C /\ E = { A } ) ) )
130	129	a1d 25	. . . . . . . . . . . . . . . . . . 19 |- ( ( A = B /\ B = C ) -> ( F = { A } -> ( E = { C } -> ( A = C /\ E = { A } ) ) ) )
131	130	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( C = D -> ( ( A = B /\ B = C ) -> ( F = { A } -> ( E = { C } -> ( A = C /\ E = { A } ) ) ) ) )
132	131	com14 96	. . . . . . . . . . . . . . . . 17 |- ( E = { C } -> ( ( A = B /\ B = C ) -> ( F = { A } -> ( C = D -> ( A = C /\ E = { A } ) ) ) ) )
133	120, 132	syl5bi 232	. . . . . . . . . . . . . . . 16 |- ( E = { C } -> ( { C } = { A , B } -> ( F = { A } -> ( C = D -> ( A = C /\ E = { A } ) ) ) ) )
134	118, 133	sylbid 230	. . . . . . . . . . . . . . 15 |- ( E = { C } -> ( E = { A , B } -> ( F = { A } -> ( C = D -> ( A = C /\ E = { A } ) ) ) ) )
135	134	com24 95	. . . . . . . . . . . . . 14 |- ( E = { C } -> ( C = D -> ( F = { A } -> ( E = { A , B } -> ( A = C /\ E = { A } ) ) ) ) )
136	135	impcom 446	. . . . . . . . . . . . 13 |- ( ( C = D /\ E = { C } ) -> ( F = { A } -> ( E = { A , B } -> ( A = C /\ E = { A } ) ) ) )
137	136	com13 88	. . . . . . . . . . . 12 |- ( E = { A , B } -> ( F = { A } -> ( ( C = D /\ E = { C } ) -> ( A = C /\ E = { A } ) ) ) )
138	137	imp 445	. . . . . . . . . . 11 |- ( ( E = { A , B } /\ F = { A } ) -> ( ( C = D /\ E = { C } ) -> ( A = C /\ E = { A } ) ) )
139	59	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( C = D /\ E = { C } ) -> ( E = { A } -> A = C ) )
140	139	com12 32	. . . . . . . . . . . . . . 15 |- ( E = { A } -> ( ( C = D /\ E = { C } ) -> A = C ) )
141	140	adantr 481	. . . . . . . . . . . . . 14 |- ( ( E = { A } /\ F = { A , B } ) -> ( ( C = D /\ E = { C } ) -> A = C ) )
142	141	imp 445	. . . . . . . . . . . . 13 |- ( ( ( E = { A } /\ F = { A , B } ) /\ ( C = D /\ E = { C } ) ) -> A = C )
143		simpl 473	. . . . . . . . . . . . . 14 |- ( ( E = { A } /\ F = { A , B } ) -> E = { A } )
144	143	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( E = { A } /\ F = { A , B } ) /\ ( C = D /\ E = { C } ) ) -> E = { A } )
145	142, 144	jca 554	. . . . . . . . . . . 12 |- ( ( ( E = { A } /\ F = { A , B } ) /\ ( C = D /\ E = { C } ) ) -> ( A = C /\ E = { A } ) )
146	145	ex 450	. . . . . . . . . . 11 |- ( ( E = { A } /\ F = { A , B } ) -> ( ( C = D /\ E = { C } ) -> ( A = C /\ E = { A } ) ) )
147	138, 146	jaoi 394	. . . . . . . . . 10 |- ( ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) -> ( ( C = D /\ E = { C } ) -> ( A = C /\ E = { A } ) ) )
148	147	impcom 446	. . . . . . . . 9 |- ( ( ( C = D /\ E = { C } ) /\ ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) ) -> ( A = C /\ E = { A } ) )
149		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( E = { A , B } -> ( E = { C } <-> { A , B } = { C } ) )
150		simpl 473	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A = B /\ B = C ) -> A = B )
151	150	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A = B /\ B = C ) /\ C = D ) -> A = B )
152	151	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A = B /\ B = C ) /\ C = D ) /\ F = { A } ) -> A = B )
153		eqtr 2641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( A = C /\ C = D ) -> A = D )
154		dfsn2 4190	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- { A } = { A , A }
155		preq2 4269	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( A = D -> { A , A } = { A , D } )
156	154, 155	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( A = D -> { A } = { A , D } )
157	153, 156	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( A = C /\ C = D ) -> { A } = { A , D } )
158	157	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( A = C -> ( C = D -> { A } = { A , D } ) )
159	121, 158	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A = B /\ B = C ) -> ( C = D -> { A } = { A , D } ) )
160	159	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( A = B /\ B = C ) /\ C = D ) -> { A } = { A , D } )
161	160	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A = B /\ B = C ) /\ C = D ) -> ( F = { A } <-> F = { A , D } ) )
162	161	biimpa 501	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A = B /\ B = C ) /\ C = D ) /\ F = { A } ) -> F = { A , D } )
163	152, 162	jca 554	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A = B /\ B = C ) /\ C = D ) /\ F = { A } ) -> ( A = B /\ F = { A , D } ) )
164	163	ex 450	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A = B /\ B = C ) /\ C = D ) -> ( F = { A } -> ( A = B /\ F = { A , D } ) ) )
165	164	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( ( A = B /\ B = C ) -> ( C = D -> ( F = { A } -> ( A = B /\ F = { A , D } ) ) ) )
166	165	com23 86	. . . . . . . . . . . . . . . . 17 |- ( ( A = B /\ B = C ) -> ( F = { A } -> ( C = D -> ( A = B /\ F = { A , D } ) ) ) )
167	99, 166	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( { A , B } = { C } -> ( F = { A } -> ( C = D -> ( A = B /\ F = { A , D } ) ) ) )
168	149, 167	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( E = { A , B } -> ( E = { C } -> ( F = { A } -> ( C = D -> ( A = B /\ F = { A , D } ) ) ) ) )
169	168	com23 86	. . . . . . . . . . . . . 14 |- ( E = { A , B } -> ( F = { A } -> ( E = { C } -> ( C = D -> ( A = B /\ F = { A , D } ) ) ) ) )
170	169	imp 445	. . . . . . . . . . . . 13 |- ( ( E = { A , B } /\ F = { A } ) -> ( E = { C } -> ( C = D -> ( A = B /\ F = { A , D } ) ) ) )
171	170	com13 88	. . . . . . . . . . . 12 |- ( C = D -> ( E = { C } -> ( ( E = { A , B } /\ F = { A } ) -> ( A = B /\ F = { A , D } ) ) ) )
172	171	imp 445	. . . . . . . . . . 11 |- ( ( C = D /\ E = { C } ) -> ( ( E = { A , B } /\ F = { A } ) -> ( A = B /\ F = { A , D } ) ) )
173	80	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( E = { A } /\ E = { C } ) -> C = A )
174	173	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( ( E = { A } /\ E = { C } ) -> ( C = D <-> A = D ) )
175	174	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( E = { A } /\ E = { C } ) -> ( C = D -> A = D ) )
176	175	ex 450	. . . . . . . . . . . . . 14 |- ( E = { A } -> ( E = { C } -> ( C = D -> A = D ) ) )
177	176	com13 88	. . . . . . . . . . . . 13 |- ( C = D -> ( E = { C } -> ( E = { A } -> A = D ) ) )
178	177	imp 445	. . . . . . . . . . . 12 |- ( ( C = D /\ E = { C } ) -> ( E = { A } -> A = D ) )
179	178	anim1d 588	. . . . . . . . . . 11 |- ( ( C = D /\ E = { C } ) -> ( ( E = { A } /\ F = { A , B } ) -> ( A = D /\ F = { A , B } ) ) )
180	172, 179	orim12d 883	. . . . . . . . . 10 |- ( ( C = D /\ E = { C } ) -> ( ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) -> ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
181	180	imp 445	. . . . . . . . 9 |- ( ( ( C = D /\ E = { C } ) /\ ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) ) -> ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) )
182	148, 181	jca 554	. . . . . . . 8 |- ( ( ( C = D /\ E = { C } ) /\ ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
183	182	ex 450	. . . . . . 7 |- ( ( C = D /\ E = { C } ) -> ( ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) ) )
184	183	com12 32	. . . . . 6 |- ( ( ( E = { A , B } /\ F = { A } ) \/ ( E = { A } /\ F = { A , B } ) ) -> ( ( C = D /\ E = { C } ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) ) )
185	184	orcoms 404	. . . . 5 |- ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) -> ( ( C = D /\ E = { C } ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) ) )
186	185	imp 445	. . . 4 |- ( ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
187	117, 186	jaoi 394	. . 3 |- ( ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) -> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )
188	55, 187	impbii 199	. 2 |- ( ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) <-> ( ( ( A = B /\ E = { A } ) /\ ( ( E = { C } /\ F = { C , D } ) \/ ( E = { C , D } /\ F = { C } ) ) ) \/ ( ( ( E = { A } /\ F = { A , B } ) \/ ( E = { A , B } /\ F = { A } ) ) /\ ( C = D /\ E = { C } ) ) ) )
189	13, 18, 188	3bitr4i 292	1 |- ( { <. A , B >. , <. C , D >. } = <. E , F >. <-> ( ( A = C /\ E = { A } ) /\ ( ( A = B /\ F = { A , D } ) \/ ( A = D /\ F = { A , B } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  propssopi  4971  fun2dmnopgexmpl  41303