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

Theorem sbctt 3342
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbctt  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )

Proof of Theorem sbctt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3282 . . . . 5  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21bibi1d 325 . . . 4  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  ph )  <->  ( [. A  /  x ]. ph  <->  ph ) ) )
32imbi2d 322 . . 3  |-  ( y  =  A  ->  (
( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )  <-> 
( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) ) )
4 sbft 2219 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
53, 4vtoclg 3119 . 2  |-  ( A  e.  V  ->  ( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) )
65imp 435 1  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455   F/wnf 1678   [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-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:  sbcgf  3343  csbtt  3386  mptsnunlem  31786
  Copyright terms: Public domain W3C validator