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Theorem cbvab 2313
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 1958 . . . 4  |-  F/ x [ z  /  y ] ps
3 cbvab.1 . . . . . 6  |-  F/ y
ph
4 cbvab.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54equcoms 1719 . . . . . . 7  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
65bicomd 141 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6sbie 1802 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
8 sbequ 1851 . . . . 5  |-  ( x  =  z  ->  ( [ x  /  y ] ps  <->  [ z  /  y ] ps ) )
97, 8bitr3id 194 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  y ] ps ) )
102, 9sbie 1802 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
11 df-clab 2176 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
12 df-clab 2176 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1310, 11, 123bitr4i 212 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1413eqriv 2186 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364   F/wnf 1471   [wsb 1773    e. wcel 2160   {cab 2175
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
This theorem is referenced by:  cbvabv  2314  cbvrab  2750  cbvsbc  3006  cbvrabcsf  3137  dfdmf  4835  dfrnf  4883  funfvdm2f  5598  abrexex2g  6140  abrexex2  6144
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