Theorem dvelimnf 2485

Description: Version of dvelim 2483 using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 2404. (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 1944	. 2 ⊢ Ⅎ𝑧𝜓
3		dvelimnf.2	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	dvelimf 2480	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is used by:  nfcvf  2951  nfrab  3453  axtcond  37017