Theorem ssopab2dv 4399

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 1924	. 2 |- ( ph -> A. x A. y ( ps -> ch ) )
3		ssopab2 4396	. 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 1396   C_ wss 3213   {copab 4172

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3219  df-ss 3226  df-opab 4174

This theorem is referenced by:  xpss12  4859  coss1  4912  coss2  4913  cnvss  4930  shftfvalg  11507  shftfval  11510  dvdsrvald  14255  dvdsrex  14260  sslm  15129  wlkex  16337