Theorem br4 31648

Description: Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)

Hypotheses

Ref	Expression
br4.1	|- ( a = A -> ( ph <-> ps ) )
br4.2	|- ( b = B -> ( ps <-> ch ) )
br4.3	|- ( c = C -> ( ch <-> th ) )
br4.4	|- ( d = D -> ( th <-> ta ) )
br4.5	|- ( x = X -> P = Q )
br4.6	|- R = { <. p , q >. | E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) }

Assertion

Ref	Expression
br4	|- ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) -> ( <. A , B >. R <. C , D >. <-> ta ) )

Distinct variable groups:   a, b, c, d, p, q, x, A   B, a, b, c, d, p, q, x   ch, b   Q, a, b, c, d, x   C, a, b, c, d, p, q, x   D, a, b, c, d, p, q, x   ps, a   X, a, b, c, d, x   P, a, b, c, d, p, q   S, a, b, c, d, p, q, x   ta, a, b, c, d, x   th, c   ph, p, q, x

Allowed substitution hints:   ph( a, b, c, d)   ps( x, q, p, b, c, d)   ch( x, q, p, a, c, d)   th( x, q, p, a, b, d)   ta( q, p)   P( x)   Q( q, p)   R( x, q, p, a, b, c, d)   X( q, p)

Proof of Theorem br4

Step	Hyp	Ref	Expression
1		opex 4932	. . 3 |- <. A , B >. e. _V
2		opex 4932	. . 3 |- <. C , D >. e. _V
3		eqeq1 2626	. . . . . . 7 |- ( p = <. A , B >. -> ( p = <. a , b >. <-> <. A , B >. = <. a , b >. ) )
4	3	3anbi1d 1403	. . . . . 6 |- ( p = <. A , B >. -> ( ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) ) )
5	4	rexbidv 3052	. . . . 5 |- ( p = <. A , B >. -> ( E. d e. P ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) ) )
6	5	2rexbidv 3057	. . . 4 |- ( p = <. A , B >. -> ( E. b e. P E. c e. P E. d e. P ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) ) )
7	6	2rexbidv 3057	. . 3 |- ( p = <. A , B >. -> ( E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) ) )
8		eqeq1 2626	. . . . . . 7 |- ( q = <. C , D >. -> ( q = <. c , d >. <-> <. C , D >. = <. c , d >. ) )
9	8	3anbi2d 1404	. . . . . 6 |- ( q = <. C , D >. -> ( ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
10	9	rexbidv 3052	. . . . 5 |- ( q = <. C , D >. -> ( E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
11	10	2rexbidv 3057	. . . 4 |- ( q = <. C , D >. -> ( E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
12	11	2rexbidv 3057	. . 3 |- ( q = <. C , D >. -> ( E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ q = <. c , d >. /\ ph ) <-> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
13		br4.6	. . 3 |- R = { <. p , q >. | E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) }
14	1, 2, 7, 12, 13	brab 4998	. 2 |- ( <. A , B >. R <. C , D >. <-> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) )
15		vex 3203	. . . . . . . . . . . 12 |- a e. _V
16		vex 3203	. . . . . . . . . . . 12 |- b e. _V
17	15, 16	opth 4945	. . . . . . . . . . 11 |- ( <. a , b >. = <. A , B >. <-> ( a = A /\ b = B ) )
18		br4.1	. . . . . . . . . . . 12 |- ( a = A -> ( ph <-> ps ) )
19		br4.2	. . . . . . . . . . . 12 |- ( b = B -> ( ps <-> ch ) )
20	18, 19	sylan9bb 736	. . . . . . . . . . 11 |- ( ( a = A /\ b = B ) -> ( ph <-> ch ) )
21	17, 20	sylbi 207	. . . . . . . . . 10 |- ( <. a , b >. = <. A , B >. -> ( ph <-> ch ) )
22	21	eqcoms 2630	. . . . . . . . 9 |- ( <. A , B >. = <. a , b >. -> ( ph <-> ch ) )
23		vex 3203	. . . . . . . . . . . 12 |- c e. _V
24		vex 3203	. . . . . . . . . . . 12 |- d e. _V
25	23, 24	opth 4945	. . . . . . . . . . 11 |- ( <. c , d >. = <. C , D >. <-> ( c = C /\ d = D ) )
26		br4.3	. . . . . . . . . . . 12 |- ( c = C -> ( ch <-> th ) )
27		br4.4	. . . . . . . . . . . 12 |- ( d = D -> ( th <-> ta ) )
28	26, 27	sylan9bb 736	. . . . . . . . . . 11 |- ( ( c = C /\ d = D ) -> ( ch <-> ta ) )
29	25, 28	sylbi 207	. . . . . . . . . 10 |- ( <. c , d >. = <. C , D >. -> ( ch <-> ta ) )
30	29	eqcoms 2630	. . . . . . . . 9 |- ( <. C , D >. = <. c , d >. -> ( ch <-> ta ) )
31	22, 30	sylan9bb 736	. . . . . . . 8 |- ( ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. ) -> ( ph <-> ta ) )
32	31	biimp3a 1432	. . . . . . 7 |- ( ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) -> ta )
33	32	a1i 11	. . . . . 6 |- ( ( ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ( x e. S /\ a e. P ) ) /\ ( b e. P /\ c e. P ) ) /\ d e. P ) -> ( ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) -> ta ) )
34	33	rexlimdva 3031	. . . . 5 |- ( ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ( x e. S /\ a e. P ) ) /\ ( b e. P /\ c e. P ) ) -> ( E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) -> ta ) )
35	34	rexlimdvva 3038	. . . 4 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ( x e. S /\ a e. P ) ) -> ( E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) -> ta ) )
36	35	rexlimdvva 3038	. . 3 |- ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) -> ( E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) -> ta ) )
37		simpl1 1064	. . . . 5 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> X e. S )
38		simpl2l 1114	. . . . . 6 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> A e. Q )
39		simpl2r 1115	. . . . . 6 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> B e. Q )
40		simpl3l 1116	. . . . . . 7 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> C e. Q )
41		simpl3r 1117	. . . . . . 7 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> D e. Q )
42		eqidd 2623	. . . . . . 7 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> <. A , B >. = <. A , B >. )
43		eqidd 2623	. . . . . . 7 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> <. C , D >. = <. C , D >. )
44		simpr 477	. . . . . . 7 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> ta )
45		opeq1 4402	. . . . . . . . . 10 |- ( c = C -> <. c , d >. = <. C , d >. )
46	45	eqeq2d 2632	. . . . . . . . 9 |- ( c = C -> ( <. C , D >. = <. c , d >. <-> <. C , D >. = <. C , d >. ) )
47	46, 26	3anbi23d 1402	. . . . . . . 8 |- ( c = C -> ( ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) <-> ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. C , d >. /\ th ) ) )
48		opeq2 4403	. . . . . . . . . 10 |- ( d = D -> <. C , d >. = <. C , D >. )
49	48	eqeq2d 2632	. . . . . . . . 9 |- ( d = D -> ( <. C , D >. = <. C , d >. <-> <. C , D >. = <. C , D >. ) )
50	49, 27	3anbi23d 1402	. . . . . . . 8 |- ( d = D -> ( ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. C , d >. /\ th ) <-> ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. C , D >. /\ ta ) ) )
51	47, 50	rspc2ev 3324	. . . . . . 7 |- ( ( C e. Q /\ D e. Q /\ ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. C , D >. /\ ta ) ) -> E. c e. Q E. d e. Q ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) )
52	40, 41, 42, 43, 44, 51	syl113anc 1338	. . . . . 6 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> E. c e. Q E. d e. Q ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) )
53		opeq1 4402	. . . . . . . . . 10 |- ( a = A -> <. a , b >. = <. A , b >. )
54	53	eqeq2d 2632	. . . . . . . . 9 |- ( a = A -> ( <. A , B >. = <. a , b >. <-> <. A , B >. = <. A , b >. ) )
55	54, 18	3anbi13d 1401	. . . . . . . 8 |- ( a = A -> ( ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> ( <. A , B >. = <. A , b >. /\ <. C , D >. = <. c , d >. /\ ps ) ) )
56	55	2rexbidv 3057	. . . . . . 7 |- ( a = A -> ( E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> E. c e. Q E. d e. Q ( <. A , B >. = <. A , b >. /\ <. C , D >. = <. c , d >. /\ ps ) ) )
57		opeq2 4403	. . . . . . . . . 10 |- ( b = B -> <. A , b >. = <. A , B >. )
58	57	eqeq2d 2632	. . . . . . . . 9 |- ( b = B -> ( <. A , B >. = <. A , b >. <-> <. A , B >. = <. A , B >. ) )
59	58, 19	3anbi13d 1401	. . . . . . . 8 |- ( b = B -> ( ( <. A , B >. = <. A , b >. /\ <. C , D >. = <. c , d >. /\ ps ) <-> ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) ) )
60	59	2rexbidv 3057	. . . . . . 7 |- ( b = B -> ( E. c e. Q E. d e. Q ( <. A , B >. = <. A , b >. /\ <. C , D >. = <. c , d >. /\ ps ) <-> E. c e. Q E. d e. Q ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) ) )
61	56, 60	rspc2ev 3324	. . . . . 6 |- ( ( A e. Q /\ B e. Q /\ E. c e. Q E. d e. Q ( <. A , B >. = <. A , B >. /\ <. C , D >. = <. c , d >. /\ ch ) ) -> E. a e. Q E. b e. Q E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) )
62	38, 39, 52, 61	syl3anc 1326	. . . . 5 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> E. a e. Q E. b e. Q E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) )
63		br4.5	. . . . . . 7 |- ( x = X -> P = Q )
64	63	rexeqdv 3145	. . . . . . . . 9 |- ( x = X -> ( E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
65	63, 64	rexeqbidv 3153	. . . . . . . 8 |- ( x = X -> ( E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
66	63, 65	rexeqbidv 3153	. . . . . . 7 |- ( x = X -> ( E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> E. b e. Q E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
67	63, 66	rexeqbidv 3153	. . . . . 6 |- ( x = X -> ( E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> E. a e. Q E. b e. Q E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
68	67	rspcev 3309	. . . . 5 |- ( ( X e. S /\ E. a e. Q E. b e. Q E. c e. Q E. d e. Q ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) -> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) )
69	37, 62, 68	syl2anc 693	. . . 4 |- ( ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) /\ ta ) -> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) )
70	69	ex 450	. . 3 |- ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) -> ( ta -> E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) ) )
71	36, 70	impbid 202	. 2 |- ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) -> ( E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P ( <. A , B >. = <. a , b >. /\ <. C , D >. = <. c , d >. /\ ph ) <-> ta ) )
72	14, 71	syl5bb 272	1 |- ( ( X e. S /\ ( A e. Q /\ B e. Q ) /\ ( C e. Q /\ D e. Q ) ) -> ( <. A , B >. R <. C , D >. <-> ta ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   {copab 4712

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  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  df-br 4654  df-opab 4713

This theorem is referenced by: (None)