Theorem bj-dvelimv 32961

Description: A version of dvelim 2368 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-dvelimv.nf	⊢ Ⅎ𝑥𝜓
bj-dvelimv.is	⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑))

Assertion

Ref	Expression
bj-dvelimv	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem bj-dvelimv

Step	Hyp	Ref	Expression
1		nfv 1883	. . 3 ⊢ Ⅎ𝑥⊤
2		bj-dvelimv.nf	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜓)
4		bj-dvelimv.is	. . 3 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑))
5	1, 3, 4	bj-dvelimdv1 32960	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑))
6	5	trud 1533	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1521  ⊤wtru 1524  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750

This theorem is referenced by:  bj-nfeel2  32962