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Theorem gencbval 2778
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 1471 . 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 233 . . . . . 6  |-  ( A  =  y  ->  (
( ch  ->  ph )  <->  ( th  ->  ps )
) )
65bicomd 140 . . . . 5  |-  ( A  =  y  ->  (
( th  ->  ps ) 
<->  ( ch  ->  ph )
) )
76eqcoms 2173 . . . 4  |-  ( y  =  A  ->  (
( th  ->  ps ) 
<->  ( ch  ->  ph )
) )
82, 7ceqsalv 2760 . . 3  |-  ( A. y ( y  =  A  ->  ( th  ->  ps ) )  <->  ( ch  ->  ph ) )
98albii 1463 . 2  |-  ( A. x A. y ( y  =  A  ->  ( th  ->  ps ) )  <->  A. x ( ch  ->  ph ) )
10 19.23v 1876 . . . 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 2172 . . . . . . . . . 10  |-  ( A  =  y  <->  y  =  A )
1312biimpi 119 . . . . . . . . 9  |-  ( A  =  y  ->  y  =  A )
1413adantl 275 . . . . . . . 8  |-  ( ( ch  /\  A  =  y )  ->  y  =  A )
1514eximi 1593 . . . . . . 7  |-  ( E. x ( ch  /\  A  =  y )  ->  E. x  y  =  A )
1611, 15sylbi 120 . . . . . 6  |-  ( th 
->  E. x  y  =  A )
17 pm2.04 82 . . . . . 6  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  -> 
( th  ->  ( E. x  y  =  A  ->  ps ) ) )
1816, 17mpdi 43 . . . . 5  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  -> 
( th  ->  ps ) )
19 ax-1 6 . . . . 5  |-  ( ( th  ->  ps )  ->  ( E. x  y  =  A  ->  ( th  ->  ps ) ) )
2018, 19impbii 125 . . . 4  |-  ( ( E. x  y  =  A  ->  ( th  ->  ps ) )  <->  ( th  ->  ps ) )
2110, 20bitri 183 . . 3  |-  ( A. x ( y  =  A  ->  ( th  ->  ps ) )  <->  ( th  ->  ps ) )
2221albii 1463 . 2  |-  ( A. y A. x ( y  =  A  ->  ( th  ->  ps ) )  <->  A. y ( th  ->  ps ) )
231, 9, 223bitr3i 209 1  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730
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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by: (None)
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