Theorem ax67to7 2168

Description: Re-derivation of ax-7 1734 from ax67 2165. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax67to7	⊢ (∀x∀yφ → ∀y∀xφ)

Proof of Theorem ax67to7

Step	Hyp	Ref	Expression
1		ax67to6 2167	. . 3 ⊢ (¬ ∀y ¬ ∀y ¬ ∀x∀yφ → ¬ ∀x∀yφ)
2	1	con4i 122	. 2 ⊢ (∀x∀yφ → ∀y ¬ ∀y ¬ ∀x∀yφ)
3		ax67 2165	. . 3 ⊢ (¬ ∀y ¬ ∀x∀yφ → ∀xφ)
4	3	alimi 1559	. 2 ⊢ (∀y ¬ ∀y ¬ ∀x∀yφ → ∀y∀xφ)
5	2, 4	syl 15	1 ⊢ (∀x∀yφ → ∀y∀xφ)

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: (None)