Theorem bj-dvelimv 34177

Description: A version of dvelim 2473 using the "nonfree" 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		bj-dvelimv.nf	. . . 4 ⊢ Ⅎ𝑥𝜓
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜓)
3		bj-dvelimv.is	. . 3 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑))
4	2, 3	bj-dvelimdv1 34176	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑))
5	4	mptru 1544	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)

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:  bj-nfeel2  34178