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Theorem cbvalivK 28161
Description: Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28136). Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
cbvalivK.1  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbvalivK  |-  ( 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 cbvalivK
StepHypRef Expression
1 ax-17 1628 . 2  |-  ( A. x ph  ->  A. y A. x ph )
2 cbvalivK.1 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32a4imvK 28151 . . 3  |-  ( A. x ph  ->  ps )
43alimiK 28142 . 2  |-  ( A. y A. x ph  ->  A. y ps )
51, 4syl 17 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  cbvalvK  28163  ax7wK  28169
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628  ax-9v 1632
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