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Theorem cbvaliw 1659
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
cbvaliw.1  |-  ( A. x ph  ->  A. y A. x ph )
cbvaliw.2  |-  ( -. 
ps  ->  A. x  -.  ps )
cbvaliw.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbvaliw  |-  ( A. x ph  ->  A. y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvaliw
StepHypRef Expression
1 cbvaliw.1 . 2  |-  ( A. x ph  ->  A. y A. x ph )
2 cbvaliw.2 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
3 cbvaliw.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3spimw 1656 . 2  |-  ( A. x ph  ->  ps )
51, 4alrimih 1555 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  cbvalw  1687
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-9 1644
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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