Theorem opelopab2 4291

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 462	. 2 |- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4	3	opelopab2a 4286	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 104   <-> wb 105   = wceq 1364   e. wcel 2160   <.cop 3613   {copab 4081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-opab 4083

This theorem is referenced by:  brecop  6655  netap  7288  2oneel  7290  2omotaplemap  7291  2omotaplemst  7292  divides  11838