ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbv2 Unicode version

Theorem cbv2 1635
Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv2.1  |-  F/ x ph
cbv2.2  |-  F/ y
ph
cbv2.3  |-  ( ph  ->  F/ y ps )
cbv2.4  |-  ( ph  ->  F/ x ch )
cbv2.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbv2  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.2 . . . 4  |-  F/ y
ph
21nfri 1412 . . 3  |-  ( ph  ->  A. y ph )
3 cbv2.1 . . . . 5  |-  F/ x ph
43nfal 1468 . . . 4  |-  F/ x A. y ph
54nfri 1412 . . 3  |-  ( A. y ph  ->  A. x A. y ph )
62, 5syl 14 . 2  |-  ( ph  ->  A. x A. y ph )
7 cbv2.3 . . . 4  |-  ( ph  ->  F/ y ps )
87nfrd 1413 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
9 cbv2.4 . . . 4  |-  ( ph  ->  F/ x ch )
109nfrd 1413 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
11 cbv2.5 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
128, 10, 11cbv2h 1634 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
136, 12syl 14 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241   F/wnf 1349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by:  cbvald  1800  cbvrald  9927
  Copyright terms: Public domain W3C validator