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Theorem cbvrab 2656
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvrab.1  |-  F/_ x A
cbvrab.2  |-  F/_ y A
cbvrab.3  |-  F/ y
ph
cbvrab.4  |-  F/ x ps
cbvrab.5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrab  |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }

Proof of Theorem cbvrab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1491 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 cbvrab.1 . . . . . 6  |-  F/_ x A
32nfcri 2250 . . . . 5  |-  F/ x  z  e.  A
4 nfs1v 1890 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfan 1527 . . . 4  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
6 eleq1 2178 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
7 sbequ12 1727 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
86, 7anbi12d 462 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
91, 5, 8cbvab 2238 . . 3  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { z  |  ( z  e.  A  /\  [ z  /  x ] ph ) }
10 cbvrab.2 . . . . . 6  |-  F/_ y A
1110nfcri 2250 . . . . 5  |-  F/ y  z  e.  A
12 cbvrab.3 . . . . . 6  |-  F/ y
ph
1312nfsb 1897 . . . . 5  |-  F/ y [ z  /  x ] ph
1411, 13nfan 1527 . . . 4  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
15 nfv 1491 . . . 4  |-  F/ z ( y  e.  A  /\  ps )
16 eleq1 2178 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
17 sbequ 1794 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
18 cbvrab.4 . . . . . . 7  |-  F/ x ps
19 cbvrab.5 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2018, 19sbie 1747 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
2117, 20syl6bb 195 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
2216, 21anbi12d 462 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
2314, 15, 22cbvab 2238 . . 3  |-  { z  |  ( z  e.  A  /\  [ z  /  x ] ph ) }  =  {
y  |  ( y  e.  A  /\  ps ) }
249, 23eqtri 2136 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { y  |  ( y  e.  A  /\  ps ) }
25 df-rab 2400 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
26 df-rab 2400 . 2  |-  { y  e.  A  |  ps }  =  { y  |  ( y  e.  A  /\  ps ) }
2724, 25, 263eqtr4i 2146 1  |-  { x  e.  A  |  ph }  =  { y  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   F/wnf 1419    e. wcel 1463   [wsb 1718   {cab 2101   F/_wnfc 2243   {crab 2395
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400
This theorem is referenced by:  cbvrabv  2657  elrabsf  2917  tfis  4465
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