Theorem cbvalvw 1969

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 → (𝜑 ↔ 𝜓))

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

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		ax-5 1839	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		ax-5 1839	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
4		ax-5 1839	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
5		cbvalvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvalw 1968	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  cbvexvw  1970  hba1w  1974  hba1wOLD  1975  ax12wdemo  2012  bj-ssbjust  32618