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Theorem cbvrald 10749
Description: Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
Hypotheses
Ref Expression
cbvrald.nf0  |-  F/ x ph
cbvrald.nf1  |-  F/ y
ph
cbvrald.nf2  |-  ( ph  ->  F/ y ps )
cbvrald.nf3  |-  ( ph  ->  F/ x ch )
cbvrald.is  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbvrald  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. y  e.  A  ch )
)
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem cbvrald
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvrald.nf0 . . . 4  |-  F/ x ph
2 nfv 1462 . . . 4  |-  F/ z
ph
3 nfv 1462 . . . . . 6  |-  F/ z  x  e.  A
43a1i 9 . . . . 5  |-  ( ph  ->  F/ z  x  e.  A )
5 nfv 1462 . . . . . 6  |-  F/ z ps
65a1i 9 . . . . 5  |-  ( ph  ->  F/ z ps )
74, 6nfimd 1518 . . . 4  |-  ( ph  ->  F/ z ( x  e.  A  ->  ps ) )
8 nfv 1462 . . . . . 6  |-  F/ x  z  e.  A
98a1i 9 . . . . 5  |-  ( ph  ->  F/ x  z  e.  A )
10 nfs1v 1857 . . . . . 6  |-  F/ x [ z  /  x ] ps
1110a1i 9 . . . . 5  |-  ( ph  ->  F/ x [ z  /  x ] ps )
129, 11nfimd 1518 . . . 4  |-  ( ph  ->  F/ x ( z  e.  A  ->  [ z  /  x ] ps ) )
13 eleq1 2142 . . . . . . 7  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
1413adantl 271 . . . . . 6  |-  ( (
ph  /\  x  =  z )  ->  (
x  e.  A  <->  z  e.  A ) )
15 sbequ12 1695 . . . . . . 7  |-  ( x  =  z  ->  ( ps 
<->  [ z  /  x ] ps ) )
1615adantl 271 . . . . . 6  |-  ( (
ph  /\  x  =  z )  ->  ( ps 
<->  [ z  /  x ] ps ) )
1714, 16imbi12d 232 . . . . 5  |-  ( (
ph  /\  x  =  z )  ->  (
( x  e.  A  ->  ps )  <->  ( z  e.  A  ->  [ z  /  x ] ps ) ) )
1817ex 113 . . . 4  |-  ( ph  ->  ( x  =  z  ->  ( ( x  e.  A  ->  ps ) 
<->  ( z  e.  A  ->  [ z  /  x ] ps ) ) ) )
191, 2, 7, 12, 18cbv2 1676 . . 3  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. z ( z  e.  A  ->  [ z  /  x ] ps ) ) )
20 cbvrald.nf1 . . . 4  |-  F/ y
ph
21 nfv 1462 . . . . . 6  |-  F/ y  z  e.  A
2221a1i 9 . . . . 5  |-  ( ph  ->  F/ y  z  e.  A )
23 cbvrald.nf2 . . . . . 6  |-  ( ph  ->  F/ y ps )
241, 23nfsbd 1893 . . . . 5  |-  ( ph  ->  F/ y [ z  /  x ] ps )
2522, 24nfimd 1518 . . . 4  |-  ( ph  ->  F/ y ( z  e.  A  ->  [ z  /  x ] ps ) )
26 nfv 1462 . . . . . 6  |-  F/ z  y  e.  A
2726a1i 9 . . . . 5  |-  ( ph  ->  F/ z  y  e.  A )
28 nfv 1462 . . . . . 6  |-  F/ z ch
2928a1i 9 . . . . 5  |-  ( ph  ->  F/ z ch )
3027, 29nfimd 1518 . . . 4  |-  ( ph  ->  F/ z ( y  e.  A  ->  ch ) )
31 eleq1 2142 . . . . . . 7  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
3231adantl 271 . . . . . 6  |-  ( (
ph  /\  z  =  y )  ->  (
z  e.  A  <->  y  e.  A ) )
33 sbequ 1762 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ps  <->  [ y  /  x ] ps ) )
34 cbvrald.nf3 . . . . . . . 8  |-  ( ph  ->  F/ x ch )
35 cbvrald.is . . . . . . . 8  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
361, 34, 35sbied 1712 . . . . . . 7  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
3733, 36sylan9bbr 451 . . . . . 6  |-  ( (
ph  /\  z  =  y )  ->  ( [ z  /  x ] ps  <->  ch ) )
3832, 37imbi12d 232 . . . . 5  |-  ( (
ph  /\  z  =  y )  ->  (
( z  e.  A  ->  [ z  /  x ] ps )  <->  ( y  e.  A  ->  ch )
) )
3938ex 113 . . . 4  |-  ( ph  ->  ( z  =  y  ->  ( ( z  e.  A  ->  [ z  /  x ] ps ) 
<->  ( y  e.  A  ->  ch ) ) ) )
402, 20, 25, 30, 39cbv2 1676 . . 3  |-  ( ph  ->  ( A. z ( z  e.  A  ->  [ z  /  x ] ps )  <->  A. y
( y  e.  A  ->  ch ) ) )
4119, 40bitrd 186 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  <->  A. y ( y  e.  A  ->  ch ) ) )
42 df-ral 2354 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
43 df-ral 2354 . 2  |-  ( A. y  e.  A  ch  <->  A. y ( y  e.  A  ->  ch )
)
4441, 42, 433bitr4g 221 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. y  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   F/wnf 1390    e. wcel 1434   [wsb 1686   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-ral 2354
This theorem is referenced by:  setindft  10918
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