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Theorem ssopab2dv 4612
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 1686 . 2  |-  ( ph  ->  A. x A. y
( ps  ->  ch ) )
3 ssopab2 4609 . 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 setvar class
Syntax hints:    -> wi 4   A.wal 1367    C_ wss 3323   {copab 4344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3330  df-ss 3337  df-opab 4346
This theorem is referenced by:  xpss12  4940  coss1  4990  coss2  4991  cnvss  5007  aceq3lem  8282  shftfval  12551  sslm  18878  ulmval  21820  iseupa  23537  fpwrelmap  25984  clwlkswlks  30376  dicssdvh  34671
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