Theorem ax467to7 2172

Description: Re-derivation of ax-7 1734 from ax467 2169. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2135) is allowed since we have already proved ax467to4 2170. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax467to7	⊢ (∀x∀yφ → ∀y∀xφ)

Proof of Theorem ax467to7

Step	Hyp	Ref	Expression
1		ax467to6 2171	. . 3 ⊢ (¬ ∀y ¬ ∀y ¬ ∀x∀yφ → ¬ ∀x∀yφ)
2	1	con4i 122	. 2 ⊢ (∀x∀yφ → ∀y ¬ ∀y ¬ ∀x∀yφ)
3		pm2.21 100	. . . . . 6 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → (∀x∀y ¬ ∀x∀yφ → ∀xφ))
4		ax467 2169	. . . . . 6 ⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ)
5	3, 4	syl 15	. . . . 5 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → φ)
6	5	alimi 1559	. . . 4 ⊢ (∀x ¬ ∀x∀y ¬ ∀x∀yφ → ∀xφ)
7		ax467to6 2171	. . . 4 ⊢ (¬ ∀x ¬ ∀x∀y ¬ ∀x∀yφ → ∀y ¬ ∀x∀yφ)
8	6, 7	nsyl4 134	. . 3 ⊢ (¬ ∀y ¬ ∀x∀yφ → ∀xφ)
9	8	alimi 1559	. 2 ⊢ (∀y ¬ ∀y ¬ ∀x∀yφ → ∀y∀xφ)
10	2, 9	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)