Theorem bj-dvelimv 37302

Description: A version of dvelim 2481 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 37301	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑))
5	4	mptru 1566	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1557  ⊤wtru 1560  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803

This theorem is referenced by:  bj-nfeel2  37303