Theorem ssopab2dv 4296

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 1886	. 2 |- ( ph -> A. x A. y ( ps -> ch ) )
3		ssopab2 4293	. 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 1362   C_ wss 3144   {copab 4078

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-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-opab 4080

This theorem is referenced by:  xpss12  4751  coss1  4800  coss2  4801  cnvss  4818  shftfvalg  10859  shftfval  10862  reldvdsrsrg  13442  dvdsrvald  13443  dvdsrex  13448  sslm  14204