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Theorem sbcbig 3324
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 3282 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  <->  ps )  <->  [. A  /  x ]. ( ph  <->  ps ) ) )
2 dfsbcq2 3282 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3282 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3bibi12d 327 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
5 sbbi 2241 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 3120 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455   [wsb 1808    e. wcel 1898   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280
This theorem is referenced by:  sbcbi1  3327  sbcabel  3357  bnj89  29577  bj-sbeq  31549  bj-sbceqgALT  31550  sbcbi  36945  sbc3orgVD  37288  sbcbiVD  37314
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