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Theorem cbvsbc 2865
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 2210 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
54eleq2i 2154 . 2  |-  ( A  e.  { x  | 
ph }  <->  A  e.  { y  |  ps }
)
6 df-sbc 2839 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
7 df-sbc 2839 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
85, 6, 73bitr4i 210 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394    e. wcel 1438   {cab 2074   [.wsbc 2838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839
This theorem is referenced by:  cbvsbcv  2866  cbvcsb  2935
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