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Theorem ssopab2dv 4375
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 1632 . 2  |-  ( ph  ->  A. x A. y
( ps  ->  ch ) )
3 ssopab2 4372 . 2  |-  ( A. x A. y ( ps 
->  ch )  ->  { <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
42, 3syl 15 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  C_  {
<. x ,  y >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1540    C_ wss 3228   {copab 4157
This theorem is referenced by:  xpss12  4874  coss1  4921  coss2  4922  cnvss  4936  aceq3lem  7837  shftfval  11661  sslm  17133  ulmval  19863  iseupa  24285  dicssdvh  31445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-in 3235  df-ss 3242  df-opab 4159
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