Theorem copsex2g 3983

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 2568	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2568	. 2 |- ( B e. W -> E. y y = B )
3		eeanv 1807	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
4		nfe1 1385	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
5		nfv 1421	. . . . 5 |- F/ x ps
6	4, 5	nfbi 1481	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
7		nfe1 1385	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
8	7	nfex 1528	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
9		nfv 1421	. . . . . 6 |- F/ y ps
10	8, 9	nfbi 1481	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
11		opeq12 3551	. . . . . . 7 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
12		copsexg 3981	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
13	12	eqcoms 2043	. . . . . . 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 179	. . . . 5 |- ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
17	10, 16	exlimi 1485	. . . 4 |- ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
18	6, 17	exlimi 1485	. . 3 |- ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
19	3, 18	sylbir 125	. 2 |- ( ( E. x x = A /\ E. y y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
20	1, 2, 19	syl2an 273	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  opelopabga  4000  ov6g  5638  ltresr  6915