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Theorem sbciegf 2873
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
sbciegf.1  |-  F/ x ps
sbciegf.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbciegf  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem sbciegf
StepHypRef Expression
1 sbciegf.1 . 2  |-  F/ x ps
2 sbciegf.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ax-gen 1384 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 sbciegft 2872 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
51, 3, 4mp3an23 1266 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1288    = wceq 1290   F/wnf 1395    e. wcel 1439   [.wsbc 2843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-sbc 2844
This theorem is referenced by:  sbcieg  2874  iunxsngf  3815  opelopabf  4112  eqerlem  6339
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