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Theorem gencbval 2992
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
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 gencbval.1 . . . 4  |-  A  e. 
_V
2 gencbval.2 . . . . 5  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
32notbid 286 . . . 4  |-  ( A  =  y  ->  ( -.  ph  <->  -.  ps )
)
4 gencbval.3 . . . 4  |-  ( A  =  y  ->  ( ch 
<->  th ) )
5 gencbval.4 . . . 4  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
61, 3, 4, 5gencbvex 2990 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  E. y
( th  /\  -.  ps ) )
7 exanali 1595 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  -.  A. x
( ch  ->  ph )
)
8 exanali 1595 . . 3  |-  ( E. y ( th  /\  -.  ps )  <->  -.  A. y
( th  ->  ps ) )
96, 7, 83bitr3i 267 . 2  |-  ( -. 
A. x ( ch 
->  ph )  <->  -.  A. y
( th  ->  ps ) )
109con4bii 289 1  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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