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

Theorem cbvalvw 1968
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 1838 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 ax-5 1838 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
3 ax-5 1838 . 2 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
4 ax-5 1838 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
5 cbvalvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvalw 1967 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704
This theorem is referenced by:  cbvexvw  1969  hba1w  1973  hba1wOLD  1974  ax12wdemo  2011  bj-ssbjust  32602
  Copyright terms: Public domain W3C validator