Theorem axc5c711 39378

Description: Proof of a single axiom that can replace ax-c5 39343, ax-c7 39345, and ax-11 2163 in a subsystem that includes these axioms plus ax-c4 39344 and ax-gen 1797 (and propositional calculus). See axc5c711toc5 39379, axc5c711toc7 39380, and axc5c711to11 39381 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 39371. (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 39343	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
2		ax10fromc7 39355	. . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3		ax-c7 39345	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑)
4	3	con1i 147	. . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
5	4	alimi 1813	. . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑)
6		ax-11 2163	. . . 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 39343	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
10	8, 9	ja 186	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Syntax hints:  ¬ wn 3  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-11 2163  ax-c5 39343  ax-c4 39344  ax-c7 39345

This theorem is referenced by:  axc5c711toc5  39379  axc5c711toc7  39380  axc5c711to11  39381