Theorem axc5c711 36859

Description: Proof of a single axiom that can replace ax-c5 36824, ax-c7 36826, and ax-11 2156 in a subsystem that includes these axioms plus ax-c4 36825 and ax-gen 1799 (and propositional calculus). See axc5c711toc5 36860, axc5c711toc7 36861, and axc5c711to11 36862 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 36852. (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 36824	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
2		ax10fromc7 36836	. . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3		ax-c7 36826	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
4	3	con1i 147	. . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5	4	alimi 1815	. . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
6		ax-11 2156	. . . 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 158	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑)
9		ax-c5 36824	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
10	8, 9	ja 186	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-11 2156  ax-c5 36824  ax-c4 36825  ax-c7 36826

This theorem is referenced by:  axc5c711toc5  36860  axc5c711toc7  36861  axc5c711to11  36862