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

Theorem cbvsbcv 3033
Description: Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
cbvsbcv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvsbcv  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Distinct variable groups:    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)    A( x, y)

Proof of Theorem cbvsbcv
StepHypRef Expression
1 nfv 1609 . 2  |-  F/ y
ph
2 nfv 1609 . 2  |-  F/ x ps
3 cbvsbcv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvsbc 3032 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [.wsbc 3004
This theorem is referenced by:  fpwwe2cbv  8268
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-sbc 3005
  Copyright terms: Public domain W3C validator