Theorem axc5c7 33214

Description: Proof of a single axiom that can replace ax-c5 33186 and ax-c7 33188. See axc5c7toc5 33215 and axc5c7toc7 33216 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c7	⊢ ((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc5c7

Step	Hyp	Ref	Expression
1		ax-c7 33188	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
2		ax-c5 33186	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3	1, 2	ja 172	1 ⊢ ((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c5 33186  ax-c7 33188

This theorem is referenced by:  axc5c7toc5  33215  axc5c7toc7  33216