MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvalw Unicode version

Theorem cbvalw 1675
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 198 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
51, 2, 4cbvaliw 1641 . 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 214 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ph ) )
98equcoms 1651 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
106, 7, 9cbvaliw 1641 . 2  |-  ( A. y ps  ->  A. x ph )
115, 10impbii 180 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  hbn1fw  1679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator