Theorem wl-cbvalnae 34305

Description: A more general version of cbval 2372 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 2427, nfsb2 2476 or dveeq1 2353. (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 1786	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1786	. . 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 34304	. 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
10	9	mptru 1529	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1520  ⊤wtru 1523  Ⅎwnf 1765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766

This theorem is referenced by: (None)