Theorem bj-dvelimv 37529

Description: A version of dvelim 2486 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 37528	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑))
5	4	mptru 1577	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ⊤wtru 1571  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  bj-nfeel2  37530