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Theorem cbvab 2601
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 2127 . . . 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 2116 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65sbbii 1713 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] ps )
72, 6bitr3i 251 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
8 df-clab 2446 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
9 df-clab 2446 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
107, 8, 93bitr4i 277 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1110eqriv 2456 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   F/wnf 1594   [wsb 1706    e. wcel 1762   {cab 2445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452
This theorem is referenced by:  cbvabv  2603  cbvrab  3104  cbvsbc  3353  cbvrabcsf  3463  rabsnifsb  4088  rabasiun  4322  dfdmf  5187  dfrnf  5232  funfv2f  5927  abrexex2g  6751  abrexex2  6755  ptrest  29476  bnj873  32936
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