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Theorem sbcbig 2921
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 2879 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  <->  ps )  <->  [. A  /  x ]. ( ph  <->  ps ) ) )
2 dfsbcq2 2879 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2879 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3bibi12d 234 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
5 sbbi 1906 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 2716 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 104    = wceq 1312    e. wcel 1461   [wsb 1716   [.wsbc 2876
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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877
This theorem is referenced by:  sbcbi1  2924  sbcabel  2956
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