Theorem wl-cbvalnae 35598

Description: A more general version of cbval 2399 when nonfree properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2449, nfsb2 2488 or dveeq1 2381. (Contributed by Wolf Lammen, 4-Jun-2019.)

Hypotheses

Ref	Expression
wl-cbvalnae.1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
wl-cbvalnae.2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
wl-cbvalnae.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
wl-cbvalnae	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem wl-cbvalnae

Step	Hyp	Ref	Expression
1		nftru 1812	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1812	. . 3 ⊢ Ⅎ𝑦⊤
3		wl-cbvalnae.1	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
4	3	a1i 11	. . 3 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑))
5		wl-cbvalnae.2	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
6	5	a1i 11	. . 3 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
7		wl-cbvalnae.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	a1i 11	. . 3 ⊢ (⊤ → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)))
9	1, 2, 4, 6, 8	wl-cbvalnaed 35597	. 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
10	9	mptru 1550	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1541  ⊤wtru 1544  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2373

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)