Theorem copsex2t 4343

Description: Closed theorem form of copsex2g 4344. (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		elisset 2818	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2818	. . . 4 |- ( B e. W -> E. y y = B )
3	1, 2	anim12i 338	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
4		eeanv 1985	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
5	3, 4	sylibr 134	. 2 |- ( ( A e. V /\ B e. W ) -> E. x E. y ( x = A /\ y = B ) )
6		nfa1 1590	. . . 4 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7		nfe1 1545	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
8		nfv 1577	. . . . 5 |- F/ x ps
9	7, 8	nfbi 1638	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
10		nfa2 1628	. . . . 5 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
11		nfe1 1545	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
12	11	nfex 1686	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
13		nfv 1577	. . . . . 6 |- F/ y ps
14	12, 13	nfbi 1638	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
15		opeq12 3869	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
16		copsexg 4342	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
17	16	eqcoms 2234	. . . . . . . . 9 |- ( <. x , y >. = <. A , B >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
18	15, 17	syl 14	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
19	18	adantl 277	. . . . . . 7 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
20		sp 1560	. . . . . . . . 9 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) )
21	20	19.21bi 1607	. . . . . . . 8 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) )
22	21	imp 124	. . . . . . 7 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( ph <-> ps ) )
23	19, 22	bitr3d 190	. . . . . 6 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) /\ ( x = A /\ y = B ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
24	23	ex 115	. . . . 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 ) ) )
25	10, 14, 24	exlimd 1646	. . . 4 |- ( 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 ) ) )
26	6, 9, 25	exlimd 1646	. . 3 |- ( 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 ) ) )
27	26	imp 124	. 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 ) )
28	5, 27	sylan2 286	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   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2202   <.cop 3676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  opelopabt  4362