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Theorem cbvalw 2010
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypotheses
Ref Expression
cbvalw.1 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
cbvalw.2 𝜓 → ∀𝑥 ¬ 𝜓)
cbvalw.3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
cbvalw.4 𝜑 → ∀𝑦 ¬ 𝜑)
cbvalw.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalw (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvalw
StepHypRef Expression
1 cbvalw.1 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 cbvalw.2 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
3 cbvalw.5 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 219 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbvaliw 1979 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
6 cbvalw.3 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
7 cbvalw.4 . . 3 𝜑 → ∀𝑦 ¬ 𝜑)
83biimprd 238 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
98equcoms 1993 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
106, 7, 9cbvaliw 1979 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
115, 10impbii 199 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  cbvalvw  2011  hbn1fw  2014
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