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Theorem cbvsbc 3032
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 2414 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
54eleq2i 2360 . 2  |-  ( A  e.  { x  | 
ph }  <->  A  e.  { y  |  ps }
)
6 df-sbc 3005 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
7 df-sbc 3005 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
85, 6, 73bitr4i 268 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534    e. wcel 1696   {cab 2282   [.wsbc 3004
This theorem is referenced by:  cbvsbcv  3033  cbvcsb  3098
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
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