Theorem opelopab2 4192

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopab2.1	|- ( x = A -> ( ph <-> ps ) )
opelopab2.2	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
opelopab2	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) )

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

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

Proof of Theorem opelopab2

Step	Hyp	Ref	Expression
1		opelopab2.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2		opelopab2.2	. . 3 |- ( y = B -> ( ps <-> ch ) )
3	1, 2	sylan9bb 457	. 2 |- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4	3	opelopab2a 4187	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   <.cop 3530   {copab 3988

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990

This theorem is referenced by:  brecop  6519  divides  11495