Theorem wl-cbvalnae 38216

Description: A more general version of cbval 2429 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 2479, nfsb2 2514 or dveeq1 2411. (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 1833	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1833	. . 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 38215	. 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
10	9	mptru 1576	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1567  ⊤wtru 1570  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813

This theorem is used by: (None)