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

Theorem cbvalivw 1681
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
cbvalivw.1  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbvalivw  |-  ( 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 cbvalivw
StepHypRef Expression
1 cbvalivw.1 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
21spimvw 1676 . 2  |-  ( A. x ph  ->  ps )
32alrimiv 1638 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546
This theorem is referenced by:  cbvalvw  1709  alcomiw  1711  ax10lem1  1969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
  Copyright terms: Public domain W3C validator