Theorem ssopab2dv 4256

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 1863	. 2 |- ( ph -> A. x A. y ( ps -> ch ) )
3		ssopab2 4253	. 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 1341   C_ wss 3116   {copab 4042

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-opab 4044

This theorem is referenced by:  xpss12  4711  coss1  4759  coss2  4760  cnvss  4777  shftfvalg  10760  shftfval  10763  sslm  12887