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Theorem ssopab2dv 4766
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 1707 . 2  |-  ( ph  ->  A. x A. y
( ps  ->  ch ) )
3 ssopab2 4763 . 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 1381    C_ wss 3461   {copab 4494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-in 3468  df-ss 3475  df-opab 4496
This theorem is referenced by:  xpss12  5098  coss1  5148  coss2  5149  cnvss  5165  aceq3lem  8504  shftfval  12882  sslm  19673  ulmval  22647  clwlkswlks  24630  iseupa  24837  fpwrelmap  27428  dicssdvh  36653  coss12d  37462
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