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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	⊢ ( ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax-c7	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 )
2		ax-c5	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )
3	1 2	ja	⊢ ( ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) → 𝜑 )