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Theorem sbcbig 3199
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 3156 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  <->  ps )  <->  [. A  /  x ]. ( ph  <->  ps ) ) )
2 dfsbcq2 3156 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3156 . . 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 2145 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 3004 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 1652   [wsb 1658    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  sbcabel  3230  sbcbi  28561  sbc3orgVD  28900  sbcbiVD  28925  bnj89  29023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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