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Theorem cbvab 2592
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
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.1 . . . . 5  |-  F/ y
ph
21sbco2 2118 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
3 cbvab.2 . . . . . 6  |-  F/ x ps
4 cbvab.3 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
53, 4sbie 2107 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65sbbii 1709 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] ps )
72, 6bitr3i 251 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
8 df-clab 2437 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
9 df-clab 2437 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
107, 8, 93bitr4i 277 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1110eqriv 2447 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   F/wnf 1590   [wsb 1702    e. wcel 1758   {cab 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443
This theorem is referenced by:  cbvabv  2594  cbvrab  3068  cbvsbc  3315  cbvrabcsf  3422  rabsnifsb  4043  dfdmf  5133  dfrnf  5178  funfv2f  5861  abrexex2g  6656  abrexex2  6660  ptrest  28565  rabasiun  30370  bnj873  32219
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