Theorem hbn1w 2080

Description: Weak version of hbn1 2178. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
hbn1w	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

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

Proof of Theorem hbn1w

Step	Hyp	Ref	Expression
1		ax-5 1942	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		ax-5 1942	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		ax-5 1942	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
4		ax-5 1942	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
5		ax-5 1942	. 2 ⊢ (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)
6		hbn1w.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	1, 2, 3, 4, 5, 6	hbn1fw 2079	1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  hba1w  2081  hbe1w  2082  ax10w  2165  wl-naevhba1v  38453