Theorem axc5c711 36932

Description: Proof of a single axiom that can replace ax-c5 36897, ax-c7 36899, and ax-11 2154 in a subsystem that includes these axioms plus ax-c4 36898 and ax-gen 1798 (and propositional calculus). See axc5c711toc5 36933, axc5c711toc7 36934, and axc5c711to11 36935 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 36925. (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 36897	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
2		ax10fromc7 36909	. . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3		ax-c7 36899	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
4	3	con1i 147	. . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5	4	alimi 1814	. . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
6		ax-11 2154	. . . 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 36897	. 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 1798  ax-4 1812  ax-11 2154  ax-c5 36897  ax-c4 36898  ax-c7 36899

This theorem is referenced by:  axc5c711toc5  36933  axc5c711toc7  36934  axc5c711to11  36935