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Theorem cbvralf 2572
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1  |-  F/_ x A
cbvralf.2  |-  F/_ y A
cbvralf.3  |-  F/ y
ph
cbvralf.4  |-  F/ x ps
cbvralf.5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvralf  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )

Proof of Theorem cbvralf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . 4  |-  F/ z ( x  e.  A  ->  ph )
2 cbvralf.1 . . . . . 6  |-  F/_ x A
32nfcri 2214 . . . . 5  |-  F/ x  z  e.  A
4 nfs1v 1857 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfim 1505 . . . 4  |-  F/ x
( z  e.  A  ->  [ z  /  x ] ph )
6 eleq1 2142 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
7 sbequ12 1695 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
86, 7imbi12d 232 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  ->  ph )  <->  ( z  e.  A  ->  [ z  /  x ] ph ) ) )
91, 5, 8cbval 1678 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. z
( z  e.  A  ->  [ z  /  x ] ph ) )
10 cbvralf.2 . . . . . 6  |-  F/_ y A
1110nfcri 2214 . . . . 5  |-  F/ y  z  e.  A
12 cbvralf.3 . . . . . 6  |-  F/ y
ph
1312nfsb 1864 . . . . 5  |-  F/ y [ z  /  x ] ph
1411, 13nfim 1505 . . . 4  |-  F/ y ( z  e.  A  ->  [ z  /  x ] ph )
15 nfv 1462 . . . 4  |-  F/ z ( y  e.  A  ->  ps )
16 eleq1 2142 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
17 sbequ 1762 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
18 cbvralf.4 . . . . . . 7  |-  F/ x ps
19 cbvralf.5 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2018, 19sbie 1715 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
2117, 20syl6bb 194 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
2216, 21imbi12d 232 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  ->  [ z  /  x ] ph )  <->  ( y  e.  A  ->  ps )
) )
2314, 15, 22cbval 1678 . . 3  |-  ( A. z ( z  e.  A  ->  [ z  /  x ] ph )  <->  A. y ( y  e.  A  ->  ps )
)
249, 23bitri 182 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. y
( y  e.  A  ->  ps ) )
25 df-ral 2354 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
26 df-ral 2354 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2724, 25, 263bitr4i 210 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   F/wnf 1390    e. wcel 1434   [wsb 1686   F/_wnfc 2207   A.wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354
This theorem is referenced by:  cbvral  2574  ffnfvf  5356
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