Theorem dvelimnf 2447

Description: Version of dvelim 2445 using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 2366. (Contributed by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)

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 1910	. 2 ⊢ Ⅎ𝑧𝜓
3		dvelimnf.2	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	dvelimf 2442	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1532  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779

This theorem is referenced by:  nfcvf  2922  nfrab  3460