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Theorem cbvsbc 3006
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 2313 . . 3  |-  { x  |  ph }  =  {
y  |  ps }
54eleq2i 2256 . 2  |-  ( A  e.  { x  | 
ph }  <->  A  e.  { y  |  ps }
)
6 df-sbc 2978 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
7 df-sbc 2978 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
85, 6, 73bitr4i 212 1  |-  ( [. A  /  x ]. ph  <->  [. A  / 
y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1471    e. wcel 2160   {cab 2175   [.wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978
This theorem is referenced by:  cbvsbcv  3007  cbvcsb  3077
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