Theorem opelopab2a 4243

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypothesis

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

Assertion

Ref	Expression
opelopab2a	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )

Distinct variable groups:   x, y, A   x, B, y   ps, x, y   x, C, y   x, D, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem opelopab2a

Step	Hyp	Ref	Expression
1		eleq1 2229	. . . . 5 |- ( x = A -> ( x e. C <-> A e. C ) )
2		eleq1 2229	. . . . 5 |- ( y = B -> ( y e. D <-> B e. D ) )
3	1, 2	bi2anan9 596	. . . 4 |- ( ( x = A /\ y = B ) -> ( ( x e. C /\ y e. D ) <-> ( A e. C /\ B e. D ) ) )
4		opelopabga.1	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
5	3, 4	anbi12d 465	. . 3 |- ( ( x = A /\ y = B ) -> ( ( ( x e. C /\ y e. D ) /\ ph ) <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
6	5	opelopabga 4241	. 2 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
7	6	bianabs 601	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   <.cop 3579   {copab 4042

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  df-opab 4044

This theorem is referenced by:  opelopab2  4248  brab2a  4657  brab2ga  4679  ltdfpr  7447