Theorem ax467 2169

Description: Proof of a single axiom that can replace ax-4 2135, ax-6o 2137, and ax-7 1734 in a subsystem that includes these axioms plus ax-5o 2136 and ax-gen 1546 (and propositional calculus). See ax467to4 2170, ax467to6 2171, and ax467to7 2172 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2162. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ax467

Step	Hyp	Ref	Expression
1		ax-4 2135	. . 3 ⊢ (∀yφ → φ)
2		ax6 2147	. . . 4 ⊢ (¬ ∀yφ → ∀y ¬ ∀yφ)
3		ax-6o 2137	. . . . . 6 ⊢ (¬ ∀x ¬ ∀x∀yφ → ∀yφ)
4	3	con1i 121	. . . . 5 ⊢ (¬ ∀yφ → ∀x ¬ ∀x∀yφ)
5	4	alimi 1559	. . . 4 ⊢ (∀y ¬ ∀yφ → ∀y∀x ¬ ∀x∀yφ)
6		ax-7 1734	. . . 4 ⊢ (∀y∀x ¬ ∀x∀yφ → ∀x∀y ¬ ∀x∀yφ)
7	2, 5, 6	3syl 18	. . 3 ⊢ (¬ ∀yφ → ∀x∀y ¬ ∀x∀yφ)
8	1, 7	nsyl4 134	. 2 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → φ)
9		ax-4 2135	. 2 ⊢ (∀xφ → φ)
10	8, 9	ja 153	1 ⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137

This theorem is referenced by:  ax467to4  2170  ax467to6  2171  ax467to7  2172