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Theorem gencbval 2658
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
Hypotheses
Ref Expression
gencbval.1  |-  A  e. 
_V
gencbval.2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
gencbval.3  |-  ( A  =  y  ->  ( ch 
<->  th ) )
gencbval.4  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
Assertion
Ref Expression
gencbval  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Distinct variable groups:    ps, x    ph, y    th, x    ch, y    y, A
Allowed substitution hints:    ph( x)    ps( y)    ch( x)    th( y)    A( x)

Proof of Theorem gencbval
StepHypRef Expression
1 alcom 1408 . 2  |-  ( A. x A. y ( y  =  A  ->  ( th  ->  ps ) )  <->  A. y A. x ( y  =  A  -> 
( th  ->  ps ) ) )
2 gencbval.1 . . . 4  |-  A  e. 
_V
3 gencbval.3 . . . . . . 7  |-  ( A  =  y  ->  ( ch 
<->  th ) )
4 gencbval.2 . . . . . . 7  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
53, 4imbi12d 232 . . . . . 6  |-  ( A  =  y  ->  (
( ch  ->  ph )  <->  ( th  ->  ps )
) )
65bicomd 139 . . . . 5  |-  ( A  =  y  ->  (
( th  ->  ps ) 
<->  ( ch  ->  ph )
) )
76eqcoms 2086 . . . 4  |-  ( y  =  A  ->  (
( th  ->  ps ) 
<->  ( ch  ->  ph )
) )
82, 7ceqsalv 2640 . . 3  |-  ( A. y ( y  =  A  ->  ( th  ->  ps ) )  <->  ( ch  ->  ph ) )
98albii 1400 . 2  |-  ( A. x A. y ( y  =  A  ->  ( th  ->  ps ) )  <->  A. x ( ch  ->  ph ) )
10 19.23v 1806 . . . 4  |-  ( A. x ( y  =  A  ->  ( th  ->  ps ) )  <->  ( E. x  y  =  A  ->  ( th  ->  ps ) ) )
11 gencbval.4 . . . . . . 7  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
12 eqcom 2085 . . . . . . . . . 10  |-  ( A  =  y  <->  y  =  A )
1312biimpi 118 . . . . . . . . 9  |-  ( A  =  y  ->  y  =  A )
1413adantl 271 . . . . . . . 8  |-  ( ( ch  /\  A  =  y )  ->  y  =  A )
1514eximi 1532 . . . . . . 7  |-  ( E. x ( ch  /\  A  =  y )  ->  E. x  y  =  A )
1611, 15sylbi 119 . . . . . 6  |-  ( th 
->  E. x  y  =  A )
17 pm2.04 81 . . . . . 6  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  -> 
( th  ->  ( E. x  y  =  A  ->  ps ) ) )
1816, 17mpdi 42 . . . . 5  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  -> 
( th  ->  ps ) )
19 ax-1 5 . . . . 5  |-  ( ( th  ->  ps )  ->  ( E. x  y  =  A  ->  ( th  ->  ps ) ) )
2018, 19impbii 124 . . . 4  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  <->  ( th  ->  ps ) )
2110, 20bitri 182 . . 3  |-  ( A. x ( y  =  A  ->  ( th  ->  ps ) )  <->  ( th  ->  ps ) )
2221albii 1400 . 2  |-  ( A. y A. x ( y  =  A  ->  ( th  ->  ps ) )  <->  A. y ( th  ->  ps ) )
231, 9, 223bitr3i 208 1  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2612
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2614
This theorem is referenced by: (None)
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