Theorem ssopab2dv 4263

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 1868	. 2 |- ( ph -> A. x A. y ( ps -> ch ) )
3		ssopab2 4260	. 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 1346   C_ wss 3121   {copab 4049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4051

This theorem is referenced by:  xpss12  4718  coss1  4766  coss2  4767  cnvss  4784  shftfvalg  10782  shftfval  10785  sslm  13041