Theorem dvelimnf 2327

Description: Version of dvelim 2325 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
dvelimnf.1	⊢ Ⅎ𝑥𝜑
dvelimnf.2	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dvelimnf	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Distinct variable group:   𝜓,𝑧

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

Proof of Theorem dvelimnf

Step	Hyp	Ref	Expression
1		dvelimnf.1	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1830	. 2 ⊢ Ⅎ𝑧𝜓
3		dvelimnf.2	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	dvelimf 2322	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  nfrab  3100