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Theorem cbvalvw 1758
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
Hypothesis
Ref Expression
cbvalvw.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvalvw  |-  ( A. x ph  <->  A. y ps )
Distinct variable groups:    x, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvalvw
StepHypRef Expression
1 ax-5 1680 . 2  |-  ( A. x ph  ->  A. y A. x ph )
2 ax-5 1680 . 2  |-  ( -. 
ps  ->  A. x  -.  ps )
3 ax-5 1680 . 2  |-  ( A. y ps  ->  A. x A. y ps )
4 ax-5 1680 . 2  |-  ( -. 
ph  ->  A. y  -.  ph )
5 cbvalvw.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvalw 1757 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  cbvexvw  1759  hba1w  1763  ax12wdemo  1780
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