Theorem opelopab2a 4048

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 2145	. . . . 5 |- ( x = A -> ( x e. C <-> A e. C ) )
2		eleq1 2145	. . . . 5 |- ( y = B -> ( y e. D <-> B e. D ) )
3	1, 2	bi2anan9 571	. . . 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 457	. . 3 |- ( ( x = A /\ y = B ) -> ( ( ( x e. C /\ y e. D ) /\ ph ) <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
6	5	opelopabga 4046	. 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 576	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 102   <-> wb 103   = wceq 1285   e. wcel 1434   <.cop 3419   {copab 3858

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860

This theorem is referenced by:  opelopab2  4053  brab2a  4439  brab2ga  4461  ltdfpr  6810