MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gencbval Structured version   Unicode version

Theorem gencbval 3152
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 292 . . . 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 3150 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  E. y
( th  /\  -.  ps ) )
7 exanali 1675 . . 3  |-  ( E. x ( ch  /\  -.  ph )  <->  -.  A. x
( ch  ->  ph )
)
8 exanali 1675 . . 3  |-  ( E. y ( th  /\  -.  ps )  <->  -.  A. y
( th  ->  ps ) )
96, 7, 83bitr3i 275 . 2  |-  ( -. 
A. x ( ch 
->  ph )  <->  -.  A. y
( th  ->  ps ) )
109con4bii 295 1  |-  ( A. x ( ch  ->  ph )  <->  A. y ( th 
->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator