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Theorem gencbval 2832
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 285 . . . 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 2830 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  E. y
( th  /\  -.  ps ) )
7 exanali 1572 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  -.  A. x
( ch  ->  ph )
)
8 exanali 1572 . . 3  |-  ( E. y ( th  /\  -.  ps )  <->  -.  A. y
( th  ->  ps ) )
96, 7, 83bitr3i 266 . 2  |-  ( -. 
A. x ( ch 
->  ph )  <->  -.  A. y
( th  ->  ps ) )
109con4bii 288 1  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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