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

Theorem cbvalh 1680
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
cbvalh.1 (𝜑 → ∀𝑦𝜑)
cbvalh.2 (𝜓 → ∀𝑥𝜓)
cbvalh.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalh (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbvalh
StepHypRef Expression
1 cbvalh.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 cbvalh.2 . . 3 (𝜓 → ∀𝑥𝜓)
3 cbvalh.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 142 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3h 1675 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63equcoms 1638 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
76biimprd 156 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3h 1675 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 124 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  cbval  1681  sb8h  1779  cbvalv  1839  sb9v  1899  sb8euh  1968
  Copyright terms: Public domain W3C validator