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Metamath Proof Explorer

Theorem axc5c7

Description: Proof of a single axiom that can replace ax-c5 and ax-c7 . See axc5c7toc5 and axc5c7toc7 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	axc5c7	$⊢ (\forall x \neg \forall x φ \to \forall x φ) \to φ$

Proof

Step	Hyp	Ref	Expression
1		ax-c7	$⊢ \neg \forall x \neg \forall x φ \to φ$
2		ax-c5	$⊢ \forall x φ \to φ$
3	1 2	ja	$⊢ (\forall x \neg \forall x φ \to \forall x φ) \to φ$