Theorem cbvalw 1955

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypotheses

Ref	Expression
cbvalw.1	⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
cbvalw.2	⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
cbvalw.3	⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
cbvalw.4	⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
cbvalw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalw	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvalw

Step	Hyp	Ref	Expression
1		cbvalw.1	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		cbvalw.2	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		cbvalw.5	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 218	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbvaliw 1920	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6		cbvalw.3	. . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
7		cbvalw.4	. . 3 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
8	3	biimprd 237	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
9	8	equcoms 1934	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
10	6, 7, 9	cbvaliw 1920	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
11	5, 10	impbii 198	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  cbvalvw  1956  hbn1fw  1959