MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbv3h Structured version   Unicode version

Theorem cbv3h 1973
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3h.1  |-  ( ph  ->  A. y ph )
cbv3h.2  |-  ( ps 
->  A. x ps )
cbv3h.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3h  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3h
StepHypRef Expression
1 cbv3h.1 . . 3  |-  ( ph  ->  A. y ph )
21nfi 1561 . 2  |-  F/ y
ph
3 cbv3h.2 . . 3  |-  ( ps 
->  A. x ps )
43nfi 1561 . 2  |-  F/ x ps
5 cbv3h.3 . 2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
62, 4, 5cbv3 1972 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem is referenced by:  cleqh  2535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
  Copyright terms: Public domain W3C validator