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

Theorem cbv3h 2265
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3h.1 (𝜑 → ∀𝑦𝜑)
cbv3h.2 (𝜓 → ∀𝑥𝜓)
cbv3h.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3h (∀𝑥𝜑 → ∀𝑦𝜓)

Proof of Theorem cbv3h
StepHypRef Expression
1 cbv3h.1 . . 3 (𝜑 → ∀𝑦𝜑)
21nf5i 2021 . 2 𝑦𝜑
3 cbv3h.2 . . 3 (𝜓 → ∀𝑥𝜓)
43nf5i 2021 . 2 𝑥𝜓
5 cbv3h.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbv3 2264 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator