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Theorem cbvsbc 3189
Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvsbc.1  |-  F/ y
ph
cbvsbc.2  |-  F/ x ps
cbvsbc.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvsbc  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )

Proof of Theorem cbvsbc
StepHypRef Expression
1 cbvsbc.1 . . . 4  |-  F/ y
ph
2 cbvsbc.2 . . . 4  |-  F/ x ps
3 cbvsbc.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvab 2554 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
54eleq2i 2500 . 2  |-  ( A  e.  { x  | 
ph }  <->  A  e.  { y  |  ps }
)
6 df-sbc 3162 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
7 df-sbc 3162 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
85, 6, 73bitr4i 269 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1553    e. wcel 1725   {cab 2422   [.wsbc 3161
This theorem is referenced by:  cbvsbcv  3190  cbvcsb  3255
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-sbc 3162
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