Theorem opelopab2a 4182

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 2200	. . . . 5 |- ( x = A -> ( x e. C <-> A e. C ) )
2		eleq1 2200	. . . . 5 |- ( y = B -> ( y e. D <-> B e. D ) )
3	1, 2	bi2anan9 595	. . . 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 464	. . 3 |- ( ( x = A /\ y = B ) -> ( ( ( x e. C /\ y e. D ) /\ ph ) <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
6	5	opelopabga 4180	. 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 600	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 1331   e. wcel 1480   <.cop 3525   {copab 3983

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985

This theorem is referenced by:  opelopab2  4187  brab2a  4587  brab2ga  4609  ltdfpr  7307