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Metamath Proof Explorer

Theorem ax10w

Description: Weak version of ax-10 from which we can prove any ax-10 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w introduced for labeling consistency. (Contributed by NM, 9-Apr-2017) Use hbn1w instead. (New usage is discouraged.)

		Ref	Expression
	Hypothesis	ax10w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	ax10w	⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax10w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	hbn1w	⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )