Theorem ax67 2165

Description: Proof of a single axiom that can replace both ax-6o 2137 and ax-7 1734. See ax67to6 2167 and ax67to7 2168 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax67	⊢ (¬ ∀x ¬ ∀y∀xφ → ∀yφ)

Proof of Theorem ax67

Step	Hyp	Ref	Expression
1		ax-7 1734	. . . . 5 ⊢ (∀y∀xφ → ∀x∀yφ)
2	1	con3i 127	. . . 4 ⊢ (¬ ∀x∀yφ → ¬ ∀y∀xφ)
3	2	alimi 1559	. . 3 ⊢ (∀x ¬ ∀x∀yφ → ∀x ¬ ∀y∀xφ)
4	3	con3i 127	. 2 ⊢ (¬ ∀x ¬ ∀y∀xφ → ¬ ∀x ¬ ∀x∀yφ)
5		ax-6o 2137	. 2 ⊢ (¬ ∀x ¬ ∀x∀yφ → ∀yφ)
6	4, 5	syl 15	1 ⊢ (¬ ∀x ¬ ∀y∀xφ → ∀yφ)

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-6o 2137

This theorem is referenced by:  ax67to6  2167  ax67to7  2168