Theorem copsex2t 4957

Description: Closed theorem form of copsex2g 4958. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
copsex2t	|- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Distinct variable groups:   x, y, ps   x, A, y   x, B, y

Allowed substitution hints:   ph( x, y)   V( x, y)   W( x, y)

Proof of Theorem copsex2t

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
2		nfe1 2027	. . . 4 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
3		nfv 1843	. . . 4 |- F/ x ps
4	2, 3	nfbi 1833	. . 3 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
5		nfa2 2040	. . . 4 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
6		nfe1 2027	. . . . . 6 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
7	6	nfex 2154	. . . . 5 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
8		nfv 1843	. . . . 5 |- F/ y ps
9	7, 8	nfbi 1833	. . . 4 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
10		opeq12 4404	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
11		copsexg 4956	. . . . . . . . 9 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
12	11	eqcoms 2630	. . . . . . . 8 |- ( <. x , y >. = <. A , B >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
13	10, 12	syl 17	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
14	13	adantl 482	. . . . . 6 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
15		2sp 2056	. . . . . . 7 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) )
16	15	imp 445	. . . . . 6 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( ph <-> ps ) )
17	14, 16	bitr3d 270	. . . . 5 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
18	17	ex 450	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) )
19	5, 9, 18	exlimd 2087	. . 3 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) )
20	1, 4, 19	exlimd 2087	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) )
21		elisset 3215	. . . 4 |- ( A e. V -> E. x x = A )
22		elisset 3215	. . . 4 |- ( B e. W -> E. y y = B )
23	21, 22	anim12i 590	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
24		eeanv 2182	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
25	23, 24	sylibr 224	. 2 |- ( ( A e. V /\ B e. W ) -> E. x E. y ( x = A /\ y = B ) )
26	20, 25	impel 485	1 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   <.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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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

This theorem is referenced by:  opelopabt  4987