Theorem ax467 1022

Description: Proof of a single axiom that can replace ax-4 972, ax-6o 977, and ax-7 961 in a subsystem that includes these axioms plus ax-5o 974 and ax-gen 962 (and propositional calculus). See ax467to4 1023, ax467to6 1024, and ax467to7 1025 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 1016.

Assertion

Ref	Expression
ax467	⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ)

Proof of Theorem ax467

Step	Hyp	Ref	Expression
1		ax-4 972	. . 3 ⊢ (∀yφ → φ)
2		hbn1 1014	. . . 4 ⊢ (¬ ∀yφ → ∀y ¬ ∀yφ)
3		ax-6o 977	. . . . . 6 ⊢ (¬ ∀x ¬ ∀x∀yφ → ∀yφ)
4	3	con1i 96	. . . . 5 ⊢ (¬ ∀yφ → ∀x ¬ ∀x∀yφ)
5	4	19.20i 991	. . . 4 ⊢ (∀y ¬ ∀yφ → ∀y∀x ¬ ∀x∀yφ)
6		ax-7 961	. . . 4 ⊢ (∀y∀x ¬ ∀x∀yφ → ∀x∀y ¬ ∀x∀yφ)
7	2, 5, 6	3syl 20	. . 3 ⊢ (¬ ∀yφ → ∀x∀y ¬ ∀x∀yφ)
8	1, 7	nsyl4 120	. 2 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → φ)
9		ax-4 972	. 2 ⊢ (∀xφ → φ)
10	8, 9	ja 137	1 ⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 953

This theorem is referenced by:  ax467to4 1023  ax467to6 1024  ax467to7 1025

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-4 972  ax-5o 974  ax-6o 977

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