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

Theorem cbvalvw 1966
 Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
Hypothesis
Ref Expression
cbvalvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalvw (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalvw
StepHypRef Expression
1 ax-5 1836 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 ax-5 1836 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
3 ax-5 1836 . 2 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
4 ax-5 1836 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
5 cbvalvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvalw 1965 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀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 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  cbvexvw  1967  hba1w  1971  hba1wOLD  1972  ax12wdemo  2009  bj-ssbjust  32313
 Copyright terms: Public domain W3C validator