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Theorem cbvexvw 1861
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 296 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
32cbvalvw 1860 . . 3  |-  ( A. x  -.  ph  <->  A. y  -.  ps )
43notbii 298 . 2  |-  ( -. 
A. x  -.  ph  <->  -. 
A. y  -.  ps )
5 df-ex 1661 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
6 df-ex 1661 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
74, 5, 63bitr4i 281 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188   A.wal 1436   E.wex 1660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840
This theorem depends on definitions:  df-bi 189  df-ex 1661
This theorem is referenced by:  suppimacnv  6934
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