MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcbig Unicode version

Theorem sbcbig 3050
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 3007 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  <->  ps )  <->  [. A  /  x ]. ( ph  <->  ps ) ) )
2 dfsbcq2 3007 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3007 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3bibi12d 312 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
5 sbbi 2024 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
61, 4, 5vtoclbg 2857 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 176    = wceq 1632   [wsb 1638    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  sbcabel  3081  sbcbi  28602  sbc3orgVD  28943  sbcbiVD  28968  bnj89  29063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator