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

Theorem cbvald 1842
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1935. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbvald.1  |-  F/ y
ph
cbvald.2  |-  ( ph  ->  F/ y ps )
cbvald.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbvald  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Distinct variable groups:    ph, x    ch, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)

Proof of Theorem cbvald
StepHypRef Expression
1 nfv 1462 . 2  |-  F/ x ph
2 cbvald.1 . 2  |-  F/ y
ph
3 cbvald.2 . 2  |-  ( ph  ->  F/ y ps )
4 nfv 1462 . . 3  |-  F/ x ch
54a1i 9 . 2  |-  ( ph  ->  F/ x ch )
6 cbvald.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
71, 2, 3, 5, 6cbv2 1676 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   F/wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbvaldva  1845
  Copyright terms: Public domain W3C validator