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Metamath Proof Explorer

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	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	hbn1w	$⊢ \neg \forall x φ \to \forall x \neg \forall x φ$

Step	Hyp	Ref	Expression
1		hbn1w.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		ax-5	$⊢ \forall x φ \to \forall y \forall x φ$
3		ax-5	$⊢ \neg ψ \to \forall x \neg ψ$
4		ax-5	$⊢ \forall y ψ \to \forall x \forall y ψ$
5		ax-5	$⊢ \neg φ \to \forall y \neg φ$
6		ax-5	$⊢ \neg \forall y ψ \to \forall x \neg \forall y ψ$
7	2 3 4 5 6 1	hbn1fw	$⊢ \neg \forall x φ \to \forall x \neg \forall x φ$