Theorem copsex2t 4274

Description: Closed theorem form of copsex2g 4275. (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 2774	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2774	. . . 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 1948	. . 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 1552	. . . 4 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7		nfe1 1507	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
8		nfv 1539	. . . . 5 |- F/ x ps
9	7, 8	nfbi 1600	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
10		nfa2 1590	. . . . 5 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
11		nfe1 1507	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
12	11	nfex 1648	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
13		nfv 1539	. . . . . 6 |- F/ y ps
14	12, 13	nfbi 1600	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
15		opeq12 3806	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
16		copsexg 4273	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
17	16	eqcoms 2196	. . . . . . . . 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 1522	. . . . . . . . 9 |- ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) -> A. y ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) )
21	20	19.21bi 1569	. . . . . . . 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 1608	. . . 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 1608	. . 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 1362   = wceq 1364   E.wex 1503   e. wcel 2164   <.cop 3621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  opelopabt  4292