Theorem onfrALTlem5VD 39121

Description: Virtual deduction proof of onfrALTlem5 38757. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 38757 is onfrALTlem5VD 39121 without virtual deductions and was automatically derived from onfrALTlem5VD 39121.

1::	|- a e. _V
2:1:	|- ( a i^i x ) e. _V
3:2:	|- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
4:3:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> -. ( a i^i x ) = (/) )
5::	|- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
6:4,5:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
7:2:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
8::	|- ( b =/= (/) <-> -. b = (/) )
9:8:	|- A. b ( b =/= (/) <-> -. b = (/) )
10:2,9:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
11:7,10:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. b =/= (/) )
12:6,11:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
13:2:	|- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
14:12,13:	|- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
15:2:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
16:15,14:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
17:2:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
18:2:	|- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
19:2:	|- [_ ( a i^i x ) / b ]_ y = y
20:18,19:	|- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
21:17,20:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
22:2:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
23:2:	|- [_ ( a i^i x ) / b ]_ (/) = (/)
24:21,23:	|- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
25:22,24:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
26:2:	|- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
27:25,26:	|- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
28:2:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
29:27,28:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
30:29:	|- A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
31:30:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
32::	|- ( E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
33:31,32:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
34:2:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
35:33,34:	|- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
36::	|- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
37:36:	|- A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
38:2,37:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
39:35,38:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
40:16,39:	|- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
41:2:	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
qed:40,41:	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem5VD	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Distinct variable groups:   a, b, y   x, b, y

Proof of Theorem onfrALTlem5VD

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- a e. _V
2	1	inex1 4799	. . 3 |- ( a i^i x ) e. _V
3		sbcimg 3477	. . 3 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) ) )
4	2, 3	e0a 38999	. 2 |- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
5		sbcangOLD 38739	. . . . 5 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) ) )
6	2, 5	e0a 38999	. . . 4 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
7		sseq1 3626	. . . . . . 7 |- ( b = ( a i^i x ) -> ( b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) ) )
8	7	sbcieg 3468	. . . . . 6 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) ) )
9	2, 8	e0a 38999	. . . . 5 |- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
10		sbcng 3476	. . . . . . . . 9 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. -. b = (/) <-> -. [. ( a i^i x ) / b ]. b = (/) ) )
11	10	bicomd 213	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) ) )
12	2, 11	e0a 38999	. . . . . . 7 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
13		df-ne 2795	. . . . . . . . 9 |- ( b =/= (/) <-> -. b = (/) )
14	13	ax-gen 1722	. . . . . . . 8 |- A. b ( b =/= (/) <-> -. b = (/) )
15		sbcbi 38749	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( A. b ( b =/= (/) <-> -. b = (/) ) -> ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) ) ) )
16	2, 14, 15	e00 38995	. . . . . . 7 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
17	12, 16	bitr4i 267	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. b =/= (/) )
18		eqsbc3 3475	. . . . . . . . 9 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) ) )
19	2, 18	e0a 38999	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
20	19	notbii 310	. . . . . . 7 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> -. ( a i^i x ) = (/) )
21		df-ne 2795	. . . . . . 7 |- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
22	20, 21	bitr4i 267	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
23	17, 22	bitr3i 266	. . . . 5 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
24	9, 23	anbi12i 733	. . . 4 |- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
25	6, 24	bitri 264	. . 3 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
26		df-rex 2918	. . . . . 6 |- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
27	26	ax-gen 1722	. . . . 5 |- A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
28		sbcbi 38749	. . . . 5 |- ( ( a i^i x ) e. _V -> ( A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) ) -> ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) ) ) )
29	2, 27, 28	e00 38995	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
30		sbcexgOLD 38753	. . . . . . 7 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) ) )
31	30	bicomd 213	. . . . . 6 |- ( ( a i^i x ) e. _V -> ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) ) )
32	2, 31	e0a 38999	. . . . 5 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
33		sbcangOLD 38739	. . . . . . . . . 10 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) ) )
34	2, 33	e0a 38999	. . . . . . . . 9 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
35		sbcel2gv 3496	. . . . . . . . . . 11 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) ) )
36	2, 35	e0a 38999	. . . . . . . . . 10 |- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
37		sbceqg 3984	. . . . . . . . . . . 12 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) ) )
38	2, 37	e0a 38999	. . . . . . . . . . 11 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
39		csbingOLD 39054	. . . . . . . . . . . . . 14 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) )
40	2, 39	e0a 38999	. . . . . . . . . . . . 13 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
41		csbvarg 4003	. . . . . . . . . . . . . . 15 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ b = ( a i^i x ) )
42	2, 41	e0a 38999	. . . . . . . . . . . . . 14 |- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
43		csbconstg 3546	. . . . . . . . . . . . . . 15 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ y = y )
44	2, 43	e0a 38999	. . . . . . . . . . . . . 14 |- [_ ( a i^i x ) / b ]_ y = y
45	42, 44	ineq12i 3812	. . . . . . . . . . . . 13 |- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
46	40, 45	eqtri 2644	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
47		csbconstg 3546	. . . . . . . . . . . . 13 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ (/) = (/) )
48	2, 47	e0a 38999	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ (/) = (/)
49	46, 48	eqeq12i 2636	. . . . . . . . . . 11 |- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
50	38, 49	bitri 264	. . . . . . . . . 10 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
51	36, 50	anbi12i 733	. . . . . . . . 9 |- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
52	34, 51	bitri 264	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
53	52	ax-gen 1722	. . . . . . 7 |- A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
54		exbi 1773	. . . . . . 7 |- ( A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ) -> ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ) )
55	53, 54	e0a 38999	. . . . . 6 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
56		df-rex 2918	. . . . . 6 |- ( E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
57	55, 56	bitr4i 267	. . . . 5 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
58	32, 57	bitr3i 266	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
59	29, 58	bitri 264	. . 3 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
60	25, 59	imbi12i 340	. 2 |- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
61	4, 60	bitri 264	1 |- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   [_csb 3533   i^i cin 3573   C_ wss 3574   (/)c0 3915

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

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-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-in 3581  df-ss 3588

This theorem is referenced by: (None)