Theorem axc5c4c711 43838

Description: Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1790 as the inference rule. This proof extends the idea of axc5c711 38390 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)

Assertion

Ref	Expression
axc5c4c711	⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))

Proof of Theorem axc5c4c711

Step	Hyp	Ref	Expression
1		axc4 2310	. . 3 ⊢ (∀𝑦(∀𝑦𝜑 → 𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
2		hbn1 2131	. . . . 5 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦 ¬ ∀𝑦(∀𝑦𝜑 → 𝜓))
3		axc7 2306	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦(∀𝑦𝜑 → 𝜓))
4	3	con1i 147	. . . . 5 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
5	2, 4	alrimih 1819	. . . 4 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
6		ax-11 2147	. . . 4 ⊢ (∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
7	5, 6	syl 17	. . 3 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
8	1, 7	nsyl4 158	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
9		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ∀𝑦𝜓))
10	9	spsd 2176	. . 3 ⊢ (¬ 𝜑 → (∀𝑦𝜑 → ∀𝑦𝜓))
11	10, 1	ja 186	. 2 ⊢ ((𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓)) → (∀𝑦𝜑 → ∀𝑦𝜓))
12	8, 11	ja 186	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-ex 1775

This theorem is referenced by:  axc5c4c711toc5  43839  axc5c4c711toc4  43840  axc5c4c711toc7  43841  axc5c4c711to11  43842