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Theorem ssopab2dv 4293
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 1618 . 2  |-  ( ph  ->  A. x A. y
( ps  ->  ch ) )
3 ssopab2 4290 . 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 1527    C_ wss 3152   {copab 4076
This theorem is referenced by:  xpss12  4792  coss1  4839  coss2  4840  cnvss  4854  aceq3lem  7747  shftfval  11565  sslm  17027  ulmval  19759  iseupa  23881  ssoprab2g  25032  dicssdvh  31376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-opab 4078
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