Theorem axc5c4c711to11 44974

Description: Rederivation of ax-11 2194 from axc5c4c711 44970. Note that ax-11 2194 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc5c4c711to11	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Proof of Theorem axc5c4c711to11

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝜑 → (∀𝑦(𝜑 → 𝜑) → 𝜑))
2	1	2alimi 1835	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))
3		axc5c4c711toc7 44973	. . . 4 ⊢ (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))
4	3	con4i 115	. . 3 ⊢ (∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))
5		pm2.21 124	. . . . . . 7 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → (∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → ∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))))
6		axc5c4c711 44970	. . . . . . . 8 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → ∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))) → (∀𝑦(𝜑 → 𝜑) → ∀𝑦𝜑))
7		sp 2221	. . . . . . . 8 ⊢ (∀𝑦𝜑 → 𝜑)
8	6, 7	syl6 36	. . . . . . 7 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ((𝜑 → 𝜑) → ∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))) → (∀𝑦(𝜑 → 𝜑) → 𝜑))
9	5, 8	syl 18	. . . . . 6 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → (∀𝑦(𝜑 → 𝜑) → 𝜑))
10	9	alimi 1834	. . . . 5 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑥(∀𝑦(𝜑 → 𝜑) → 𝜑))
11		axc5c4c711toc7 44973	. . . . 5 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑))
12	10, 11	nsyl4 159	. . . 4 ⊢ (¬ ∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑥(∀𝑦(𝜑 → 𝜑) → 𝜑))
13	12	alimi 1834	. . 3 ⊢ (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑦∀𝑥(∀𝑦(𝜑 → 𝜑) → 𝜑))
14	4, 13	syl 18	. 2 ⊢ (∀𝑥∀𝑦(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑦∀𝑥(∀𝑦(𝜑 → 𝜑) → 𝜑))
15		pm2.27 43	. . . 4 ⊢ (∀𝑦(𝜑 → 𝜑) → ((∀𝑦(𝜑 → 𝜑) → 𝜑) → 𝜑))
16		id 23	. . . 4 ⊢ (𝜑 → 𝜑)
17	15, 16	mpg 1820	. . 3 ⊢ ((∀𝑦(𝜑 → 𝜑) → 𝜑) → 𝜑)
18	17	2alimi 1835	. 2 ⊢ (∀𝑦∀𝑥(∀𝑦(𝜑 → 𝜑) → 𝜑) → ∀𝑦∀𝑥𝜑)
19	2, 14, 18	3syl 19	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by: (None)