Theorem opelopabt 4987

Description: Closed theorem form of opelopab 4997. (Contributed by NM, 19-Feb-2013.)

Assertion

Ref	Expression
opelopabt	|- ( ( A. x A. y ( x = A -> ( ph <-> ps ) ) /\ A. x A. y ( y = B -> ( ps <-> ch ) ) /\ ( A e. V /\ B e. W ) ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Distinct variable groups:   x, y, A   x, B, y   ch, x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   V( x, y)   W( x, y)

Proof of Theorem opelopabt

Step	Hyp	Ref	Expression
1		elopab 4983	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) )
2		19.26-2 1799	. . . 4 |- ( A. x A. y ( ( x = A -> ( ph <-> ps ) ) /\ ( y = B -> ( ps <-> ch ) ) ) <-> ( A. x A. y ( x = A -> ( ph <-> ps ) ) /\ A. x A. y ( y = B -> ( ps <-> ch ) ) ) )
3		prth 595	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ( y = B -> ( ps <-> ch ) ) ) -> ( ( x = A /\ y = B ) -> ( ( ph <-> ps ) /\ ( ps <-> ch ) ) ) )
4		bitr 745	. . . . . 6 |- ( ( ( ph <-> ps ) /\ ( ps <-> ch ) ) -> ( ph <-> ch ) )
5	3, 4	syl6 35	. . . . 5 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ( y = B -> ( ps <-> ch ) ) ) -> ( ( x = A /\ y = B ) -> ( ph <-> ch ) ) )
6	5	2alimi 1740	. . . 4 |- ( A. x A. y ( ( x = A -> ( ph <-> ps ) ) /\ ( y = B -> ( ps <-> ch ) ) ) -> A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ch ) ) )
7	2, 6	sylbir 225	. . 3 |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) ) /\ A. x A. y ( y = B -> ( ps <-> ch ) ) ) -> A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ch ) ) )
8		copsex2t 4957	. . 3 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> ( ph <-> ch ) ) /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ch ) )
9	7, 8	stoic3 1701	. 2 |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) ) /\ A. x A. y ( y = B -> ( ps <-> ch ) ) /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ch ) )
10	1, 9	syl5bb 272	1 |- ( ( A. x A. y ( x = A -> ( ph <-> ps ) ) /\ A. x A. y ( y = B -> ( ps <-> ch ) ) /\ ( A e. V /\ B e. W ) ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by: (None)