Theorem copsex2t 4331

Description: Closed theorem form of copsex2g 4332. (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 2814	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2814	. . . 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 1983	. . 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 1587	. . . 4 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7		nfe1 1542	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
8		nfv 1574	. . . . 5 |- F/ x ps
9	7, 8	nfbi 1635	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
10		nfa2 1625	. . . . 5 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
11		nfe1 1542	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
12	11	nfex 1683	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
13		nfv 1574	. . . . . 6 |- F/ y ps
14	12, 13	nfbi 1635	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
15		opeq12 3859	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
16		copsexg 4330	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
17	16	eqcoms 2232	. . . . . . . . 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 1557	. . . . . . . . 9 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) )
21	20	19.21bi 1604	. . . . . . . 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 1643	. . . 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 1643	. . 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 1393   = wceq 1395   E.wex 1538   e. wcel 2200   <.cop 3669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  opelopabt  4350