Theorem copsex2g 4231

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 2744	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2744	. 2 |- ( B e. W -> E. y y = B )
3		eeanv 1925	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
4		nfe1 1489	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
5		nfv 1521	. . . . 5 |- F/ x ps
6	4, 5	nfbi 1582	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
7		nfe1 1489	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
8	7	nfex 1630	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
9		nfv 1521	. . . . . 6 |- F/ y ps
10	8, 9	nfbi 1582	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
11		opeq12 3767	. . . . . . 7 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
12		copsexg 4229	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
13	12	eqcoms 2173	. . . . . . 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 1587	. . . 4 |- ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
18	6, 17	exlimi 1587	. . 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 1348   E.wex 1485   e. wcel 2141   <.cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  opelopabga  4248  ov6g  5990  ltresr  7801