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Theorem cbvalvw 1709
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
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 cbvalvw.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21biimpd 199 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32cbvalivw 1681 . 2  |-  ( A. x ph  ->  A. y ps )
41biimprd 215 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ph ) )
54equcoms 1688 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
65cbvalivw 1681 . 2  |-  ( A. y ps  ->  A. x ph )
73, 6impbii 181 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546
This theorem is referenced by:  cbvexvw  1710  hba1w  1714  ax11wdemo  1730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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