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Theorem cbvabw 2293
Description: Version of cbvab 2294 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Hypotheses
Ref Expression
cbvabw.1  |-  F/ y
ph
cbvabw.2  |-  F/ x ps
cbvabw.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvabw  |-  { x  |  ph }  =  {
y  |  ps }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvabw
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvabw.1 . . . . . 6  |-  F/ y
ph
21nfsbv 1940 . . . . 5  |-  F/ y [ z  /  x ] ph
3 equequ2 1706 . . . . . . . 8  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
43imbi1d 230 . . . . . . 7  |-  ( y  =  z  ->  (
( x  =  y  ->  ph )  <->  ( x  =  z  ->  ph )
) )
54albidv 1817 . . . . . 6  |-  ( y  =  z  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  z  ->  ph ) ) )
6 sb6 1879 . . . . . 6  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
7 sb6 1879 . . . . . 6  |-  ( [ z  /  x ] ph 
<-> 
A. x ( x  =  z  ->  ph )
)
85, 6, 73bitr4g 222 . . . . 5  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
92, 8sbiev 1785 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] ph )
10 cbvabw.2 . . . . . 6  |-  F/ x ps
11 cbvabw.3 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1210, 11sbiev 1785 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
1312sbbii 1758 . . . 4  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] ps )
149, 13bitr3i 185 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
15 df-clab 2157 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
16 df-clab 2157 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1714, 15, 163bitr4i 211 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1817eqriv 2167 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346    = wceq 1348   F/wnf 1453   [wsb 1755    e. wcel 2141   {cab 2156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163
This theorem is referenced by:  cbvsbcw  2982
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