Theorem wl-cbvalnae 34775

Description: A more general version of cbval 2416 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2470, nfsb2 2522 or dveeq1 2398. (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 1805	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1805	. . 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 34774	. 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
10	9	mptru 1544	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1535  ⊤wtru 1538  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)