Theorem hbn1w

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

	Ref	Expression
Hypothesis	hbn1w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	hbn1w	⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )

Proof

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