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Theorem ssopab2dv 4486
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
StepHypRef Expression
1 ssopab2dv.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21alrimivv 1643 . 2  |-  ( ph  ->  A. x A. y
( ps  ->  ch ) )
3 ssopab2 4483 . 2  |-  ( A. x A. y ( ps 
->  ch )  ->  { <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
42, 3syl 16 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  C_  {
<. x ,  y >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    C_ wss 3322   {copab 4268
This theorem is referenced by:  xpss12  4984  coss1  5031  coss2  5032  cnvss  5048  aceq3lem  8006  shftfval  11890  sslm  17368  ulmval  20301  iseupa  21692  dicssdvh  32058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336  df-opab 4270
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