Theorem copsex2g 4224

Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)

Hypothesis

Ref	Expression
copsex2g.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
copsex2g	|- ( ( 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 copsex2g

Step	Hyp	Ref	Expression
1		elisset 2740	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2740	. 2 |- ( B e. W -> E. y y = B )
3		eeanv 1920	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
4		nfe1 1484	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
5		nfv 1516	. . . . 5 |- F/ x ps
6	4, 5	nfbi 1577	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
7		nfe1 1484	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
8	7	nfex 1625	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
9		nfv 1516	. . . . . 6 |- F/ y ps
10	8, 9	nfbi 1577	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
11		opeq12 3760	. . . . . . 7 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
12		copsexg 4222	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
13	12	eqcoms 2168	. . . . . . 7 |- ( <. x , y >. = <. A , B >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
14	11, 13	syl 14	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
15		copsex2g.1	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
16	14, 15	bitr3d 189	. . . . 5 |- ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
17	10, 16	exlimi 1582	. . . 4 |- ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
18	6, 17	exlimi 1582	. . 3 |- ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
19	3, 18	sylbir 134	. 2 |- ( ( E. x x = A /\ E. y y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
20	1, 2, 19	syl2an 287	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   <.cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  opelopabga  4241  ov6g  5979  ltresr  7780