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Theorem cbvalw 1715
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypotheses
Ref Expression
cbvalw.1  |-  ( A. x ph  ->  A. y A. x ph )
cbvalw.2  |-  ( -. 
ps  ->  A. x  -.  ps )
cbvalw.3  |-  ( A. y ps  ->  A. x A. y ps )
cbvalw.4  |-  ( -. 
ph  ->  A. y  -.  ph )
cbvalw.5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvalw  |-  ( A. x ph  <->  A. y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvalw
StepHypRef Expression
1 cbvalw.1 . . 3  |-  ( A. x ph  ->  A. y A. x ph )
2 cbvalw.2 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
3 cbvalw.5 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43biimpd 200 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
51, 2, 4cbvaliw 1686 . 2  |-  ( A. x ph  ->  A. y ps )
6 cbvalw.3 . . 3  |-  ( A. y ps  ->  A. x A. y ps )
7 cbvalw.4 . . 3  |-  ( -. 
ph  ->  A. y  -.  ph )
83biimprd 216 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ph ) )
98equcoms 1694 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
106, 7, 9cbvaliw 1686 . 2  |-  ( A. y ps  ->  A. x ph )
115, 10impbii 182 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  cbvalvw  1716  hbn1fw  1720  hbn1fwOLD  1721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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