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Theorem cbvalvK 27916
Description: Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 27889). (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
cbvalvK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvalvK  |-  ( 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 cbvalvK
StepHypRef Expression
1 cbvalvK.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21biimpd 200 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32cbvalivK 27914 . 2  |-  ( A. x ph  ->  A. y ps )
4 equcomiK 27890 . . . 4  |-  ( y  =  x  ->  x  =  y )
51biimprd 216 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ph ) )
64, 5syl 17 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
76cbvalivK 27914 . 2  |-  ( A. y ps  ->  A. x ph )
83, 7impbii 182 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532
This theorem is referenced by:  cbvexvK  27917  ax6wK  27919  hba1wK  27920  ax11wdemoK  27929  ax12o10lem5K  27942  ax12o10lem8K  27948  ax12olem20K  27973
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-8 1623  ax-17 1628  ax-9v 1632
This theorem depends on definitions:  df-bi 179
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