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Theorem sbcbig 3150
Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
Assertion
Ref Expression
sbcbig  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )

Proof of Theorem sbcbig
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3107 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  <->  ps )  <->  [. A  /  x ]. ( ph  <->  ps ) ) )
2 dfsbcq2 3107 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3107 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3bibi12d 313 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
5 sbbi 2104 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 2955 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   [wsb 1655    e. wcel 1717   [.wsbc 3104
This theorem is referenced by:  sbcabel  3181  sbcbi  27967  sbc3orgVD  28304  sbcbiVD  28329  bnj89  28424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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