Theorem axc5c711 34203

Description: Proof of a single axiom that can replace ax-c5 34168, ax-c7 34170, and ax-11 2034 in a subsystem that includes these axioms plus ax-c4 34169 and ax-gen 1722 (and propositional calculus). See axc5c711toc5 34204, axc5c711toc7 34205, and axc5c711to11 34206 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 34196. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c711	⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc5c711

Step	Hyp	Ref	Expression
1		ax-c5 34168	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
2		ax10fromc7 34180	. . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3		ax-c7 34170	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
4	3	con1i 144	. . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5	4	alimi 1739	. . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
6		ax-11 2034	. . . 4 ⊢ (∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
7	2, 5, 6	3syl 18	. . 3 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑)
8	1, 7	nsyl4 156	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑)
9		ax-c5 34168	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
10	8, 9	ja 173	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-11 2034  ax-c5 34168  ax-c4 34169  ax-c7 34170

This theorem is referenced by:  axc5c711toc5  34204  axc5c711toc7  34205  axc5c711to11  34206