Theorem ssopab2dv 4033

Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypothesis

Ref	Expression
ssopab2dv.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
ssopab2dv	|- ( ph -> { <. x , y >. | ps } C_ { <. x , y >. | ch } )

Distinct variable groups:   ph, x   ph, y

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

Proof of Theorem ssopab2dv

Step	Hyp	Ref	Expression
1		ssopab2dv.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	alrimivv 1796	. 2 |- ( ph -> A. x A. y ( ps -> ch ) )
3		ssopab2 4030	. 2 |- ( A. x A. y ( ps -> ch ) -> { <. x , y >. | ps } C_ { <. x , y >. | ch } )
4	2, 3	syl 14	1 |- ( ph -> { <. x , y >. | ps } C_ { <. x , y >. | ch } )

Syntax hints:   -> wi 4   A.wal 1282   C_ wss 2973   {copab 3838

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-opab 3840

This theorem is referenced by:  xpss12  4463  coss1  4509  coss2  4510  cnvss  4526  shftfvalg  9706  shftfval  9709