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Theorem cbvralf 3082
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 1683 . . . 4  |-  F/ z ( x  e.  A  ->  ph )
2 cbvralf.1 . . . . . 6  |-  F/_ x A
32nfcri 2622 . . . . 5  |-  F/ x  z  e.  A
4 nfs1v 2164 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfim 1867 . . . 4  |-  F/ x
( z  e.  A  ->  [ z  /  x ] ph )
6 eleq1 2539 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
7 sbequ12 1961 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
86, 7imbi12d 320 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  ->  ph )  <->  ( z  e.  A  ->  [ z  /  x ] ph ) ) )
91, 5, 8cbval 1994 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. z
( z  e.  A  ->  [ z  /  x ] ph ) )
10 cbvralf.2 . . . . . 6  |-  F/_ y A
1110nfcri 2622 . . . . 5  |-  F/ y  z  e.  A
12 cbvralf.3 . . . . . 6  |-  F/ y
ph
1312nfsb 2168 . . . . 5  |-  F/ y [ z  /  x ] ph
1411, 13nfim 1867 . . . 4  |-  F/ y ( z  e.  A  ->  [ z  /  x ] ph )
15 nfv 1683 . . . 4  |-  F/ z ( y  e.  A  ->  ps )
16 eleq1 2539 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
17 sbequ 2090 . . . . . 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 2123 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
2117, 20syl6bb 261 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
2216, 21imbi12d 320 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  ->  [ z  /  x ] ph )  <->  ( y  e.  A  ->  ps )
) )
2314, 15, 22cbval 1994 . . 3  |-  ( A. z ( z  e.  A  ->  [ z  /  x ] ph )  <->  A. y ( y  e.  A  ->  ps )
)
249, 23bitri 249 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. y
( y  e.  A  ->  ps ) )
25 df-ral 2819 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
26 df-ral 2819 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2724, 25, 263bitr4i 277 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377   F/wnf 1599   [wsb 1711    e. wcel 1767   F/_wnfc 2615   A.wral 2814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819
This theorem is referenced by:  cbvrexf  3083  cbvral  3084  reusv2lem4  4651  reusv2  4653  ffnfvf  6049  nnwof  11149  nnindf  27375  scottexf  30407  scott0f  30408  evth2f  31195  evthf  31207  stoweidlem14  31541  stoweidlem28  31555  stoweidlem59  31586
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