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Theorem cbvab 2543
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 2182 . . . 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 2173 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
65sbbii 1770 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] ps )
72, 6bitr3i 251 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
8 df-clab 2388 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
9 df-clab 2388 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
107, 8, 93bitr4i 277 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1110eqriv 2398 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405   F/wnf 1637   [wsb 1763    e. wcel 1842   {cab 2387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394
This theorem is referenced by:  cbvabv  2545  cbvrab  3057  cbvsbc  3306  cbvrabcsf  3408  rabsnifsb  4040  rabasiun  4275  dfdmf  5017  dfrnf  5062  funfv2f  5918  abrexex2g  6761  abrexex2  6765  bnj873  29309  ptrest  31420
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