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Theorem cbvsbc 2932
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 2261 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
54eleq2i 2204 . 2  |-  ( A  e.  { x  | 
ph }  <->  A  e.  { y  |  ps }
)
6 df-sbc 2905 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
7 df-sbc 2905 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
85, 6, 73bitr4i 211 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1436    e. wcel 1480   {cab 2123   [.wsbc 2904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905
This theorem is referenced by:  cbvsbcv  2933  cbvcsb  3003
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