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Theorem cbvab 2401
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 2048 . . . 4  |-  F/ x [ z  /  y ] ps
3 cbvab.1 . . . . . 6  |-  F/ y
ph
4 cbvab.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54equcoms 1651 . . . . . . 7  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
65bicomd 192 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6sbie 1978 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
8 sbequ 2000 . . . . 5  |-  ( x  =  z  ->  ( [ x  /  y ] ps  <->  [ z  /  y ] ps ) )
97, 8syl5bbr 250 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  y ] ps ) )
102, 9sbie 1978 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
11 df-clab 2270 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
12 df-clab 2270 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1310, 11, 123bitr4i 268 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1413eqriv 2280 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269
This theorem is referenced by:  cbvabv  2402  cbvrab  2786  cbvsbc  3019  cbvrabcsf  3146  dfdmf  4873  dfrnf  4917  funfv2f  5588  abrexex2g  5768  abrexex2  5780  bnj873  28956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276
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