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Theorem sbcbi1 2930
Description: Distribution of class substitution over biconditional. One direction of sbcbig 2927 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcbi1  |-  ( [. A  /  x ]. ( ph 
<->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) )

Proof of Theorem sbcbi1
StepHypRef Expression
1 sbcex 2890 . 2  |-  ( [. A  /  x ]. ( ph 
<->  ps )  ->  A  e.  _V )
2 sbcbig 2927 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
32biimpd 143 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
41, 3mpcom 36 1  |-  ( [. A  /  x ]. ( ph 
<->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1465   _Vcvv 2660   [.wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by: (None)
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