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Theorem sbciegf 3331
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 1665 . 2  |-  A. x
( x  =  A  ->  ( ph  <->  ps )
)
4 sbciegft 3330 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
51, 3, 4mp3an23 1352 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   F/wnf 1663    e. wcel 1868   [.wsbc 3299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3083  df-sbc 3300
This theorem is referenced by:  sbcieg  3332  opelopabgf  4737  opelopabf  4742  eqerlem  7400  iunxsngf  28162  bnj919  29574  bnj1464  29651  bnj1123  29791  bnj1373  29835  poimirlem25  31879  sbccomieg  35555  aomclem6  35837  fveqsb  36664  rexsngf  37253
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