Theorem axc5c711 38874

Description: Proof of a single axiom that can replace ax-c5 38839, ax-c7 38841, and ax-11 2158 in a subsystem that includes these axioms plus ax-c4 38840 and ax-gen 1793 (and propositional calculus). See axc5c711toc5 38875, axc5c711toc7 38876, and axc5c711to11 38877 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 38867. (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 38839	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
2		ax10fromc7 38851	. . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3		ax-c7 38841	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
4	3	con1i 147	. . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5	4	alimi 1809	. . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
6		ax-11 2158	. . . 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 38839	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
10	8, 9	ja 186	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-11 2158  ax-c5 38839  ax-c4 38840  ax-c7 38841

This theorem is referenced by:  axc5c711toc5  38875  axc5c711toc7  38876  axc5c711to11  38877