Theorem copsex2g 4308

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 2791	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2791	. 2 |- ( B e. W -> E. y y = B )
3		eeanv 1961	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
4		nfe1 1520	. . . . 5 |- F/ x E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
5		nfv 1552	. . . . 5 |- F/ x ps
6	4, 5	nfbi 1613	. . . 4 |- F/ x ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
7		nfe1 1520	. . . . . . 7 |- F/ y E. y ( <. A , B >. = <. x , y >. /\ ph )
8	7	nfex 1661	. . . . . 6 |- F/ y E. x E. y ( <. A , B >. = <. x , y >. /\ ph )
9		nfv 1552	. . . . . 6 |- F/ y ps
10	8, 9	nfbi 1613	. . . . 5 |- F/ y ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps )
11		opeq12 3835	. . . . . . 7 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
12		copsexg 4306	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( ph <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) ) )
13	12	eqcoms 2210	. . . . . . 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 190	. . . . 5 |- ( ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
17	10, 16	exlimi 1618	. . . 4 |- ( E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
18	6, 17	exlimi 1618	. . 3 |- ( E. x E. y ( x = A /\ y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
19	3, 18	sylbir 135	. 2 |- ( ( E. x x = A /\ E. y y = B ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
20	1, 2, 19	syl2an 289	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   <.cop 3646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  opelopabga  4327  ov6g  6107  ltresr  7987