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Theorem cbvalvw 1688
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 198 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32cbvalivw 1660 . 2  |-  ( A. x ph  ->  A. y ps )
41biimprd 214 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ph ) )
54equcoms 1666 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
65cbvalivw 1660 . 2  |-  ( A. y ps  ->  A. x ph )
73, 6impbii 180 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  cbvexvw  1689  hba1w  1693  ax11wdemo  1709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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