Theorem axc5c4c711 38602

Description: Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1722 as the inference rule. This proof extends the idea of axc5c711 34203 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 2130	. . 3 ⊢ (∀𝑦(∀𝑦𝜑 → 𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
2		hbn1 2020	. . . . 5 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦 ¬ ∀𝑦(∀𝑦𝜑 → 𝜓))
3		axc7 2132	. . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦(∀𝑦𝜑 → 𝜓))
4	3	con1i 144	. . . . 5 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
5	2, 4	alrimih 1751	. . . 4 ⊢ (¬ ∀𝑦(∀𝑦𝜑 → 𝜓) → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓))
6		ax-11 2034	. . . 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 156	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (∀𝑦𝜑 → ∀𝑦𝜓))
9		pm2.21 120	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ∀𝑦𝜓))
10	9	spsd 2057	. . 3 ⊢ (¬ 𝜑 → (∀𝑦𝜑 → ∀𝑦𝜓))
11	10, 1	ja 173	. 2 ⊢ ((𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓)) → (∀𝑦𝜑 → ∀𝑦𝜓))
12	8, 11	ja 173	1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  axc5c4c711toc5  38603  axc5c4c711toc4  38604  axc5c4c711toc7  38605  axc5c4c711to11  38606