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Theorem cbvab 2263
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
cbvab.1  |-  F/ y
ph
cbvab.2  |-  F/ x ps
cbvab.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvab  |-  { x  |  ph }  =  {
y  |  ps }

Proof of Theorem cbvab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5  |-  F/ x ps
21nfsb 1919 . . . 4  |-  F/ x [ z  /  y ] ps
3 cbvab.1 . . . . . 6  |-  F/ y
ph
4 cbvab.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54equcoms 1684 . . . . . . 7  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
65bicomd 140 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6sbie 1764 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
8 sbequ 1812 . . . . 5  |-  ( x  =  z  ->  ( [ x  /  y ] ps  <->  [ z  /  y ] ps ) )
97, 8bitr3id 193 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  y ] ps ) )
102, 9sbie 1764 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
11 df-clab 2126 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
12 df-clab 2126 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1310, 11, 123bitr4i 211 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1413eqriv 2136 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   F/wnf 1436    e. wcel 1480   [wsb 1735   {cab 2125
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 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132
This theorem is referenced by:  cbvabv  2264  cbvrab  2684  cbvsbc  2937  cbvrabcsf  3065  dfdmf  4732  dfrnf  4780  funfvdm2f  5486  abrexex2g  6018  abrexex2  6022
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