MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbv3 Structured version   Unicode version

Theorem cbv3 1977
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 2207. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3.1  |-  F/ y
ph
cbv3.2  |-  F/ x ps
cbv3.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . 3  |-  F/ y
ph
21nfal 1889 . 2  |-  F/ y A. x ph
3 cbv3.2 . . 3  |-  F/ x ps
4 cbv3.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4spim 1968 . 2  |-  ( A. x ph  ->  ps )
62, 5alrimi 1820 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1372   F/wnf 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595
This theorem is referenced by:  cbv3h  1978  cbv1  1979  cbval  1987  axc16i  2030  mo3OLD  2314  moOLD  2320  axc11next  30710
  Copyright terms: Public domain W3C validator