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Theorem cbvexvw 1689
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
Hypothesis
Ref Expression
cbvalvw.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvexvw  |-  ( E. x ph  <->  E. y ps )
Distinct variable groups:    x, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvexvw
StepHypRef Expression
1 cbvalvw.1 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32cbvalvw 1688 . . 3  |-  ( A. x  -.  ph  <->  A. y  -.  ps )
43notbii 287 . 2  |-  ( -. 
A. x  -.  ph  <->  -. 
A. y  -.  ps )
5 df-ex 1532 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1532 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4i 268 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531
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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