Theorem axc5c4c711toc7 42022

Description: Rederivation of axc7 2311 from axc5c4c711 42019. Note that neither axc7 2311 nor ax-11 2154 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c4c711toc7	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Proof of Theorem axc5c4c711toc7

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝜑) → 𝜑))
2	1	alimi 1814	. . . . . . 7 ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑))
3	2	axc4i 2316	. . . . . 6 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑))
4	3	con3i 154	. . . . 5 ⊢ (¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → ¬ ∀𝑥𝜑)
5	4	alimi 1814	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
6	5	sps 2178	. . 3 ⊢ (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → ∀𝑥 ¬ ∀𝑥𝜑)
7	6	con3i 154	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑))
8		pm2.21 123	. . . 4 ⊢ (¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → ∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑))))
9		axc5c4c711 42019	. . . 4 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → ∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑))) → (∀𝑥(𝜑 → 𝜑) → ∀𝑥𝜑))
10	8, 9	syl 17	. . 3 ⊢ (¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → (∀𝑥(𝜑 → 𝜑) → ∀𝑥𝜑))
11		sp 2176	. . 3 ⊢ (∀𝑥𝜑 → 𝜑)
12	10, 11	syl6 35	. 2 ⊢ (¬ ∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥(𝜑 → 𝜑) → 𝜑) → (∀𝑥(𝜑 → 𝜑) → 𝜑))
13		pm2.27 42	. . 3 ⊢ (∀𝑥(𝜑 → 𝜑) → ((∀𝑥(𝜑 → 𝜑) → 𝜑) → 𝜑))
14		id 22	. . 3 ⊢ (𝜑 → 𝜑)
15	13, 14	mpg 1800	. 2 ⊢ ((∀𝑥(𝜑 → 𝜑) → 𝜑) → 𝜑)
16	7, 12, 15	3syl 18	1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀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-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  axc5c4c711to11  42023