Theorem ax467to7 1025

Description: Re-derivation of ax-7 961 from ax467 1022. Note that ax-6o 977 and ax-7 961 are not used by the re-derivation. The use of 19.20i 991 (which uses ax-4 972) is allowed since we have already proved ax467to4 1023.

Assertion

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

Proof of Theorem ax467to7

Step	Hyp	Ref	Expression
1		ax467to6 1024	. . 3 ⊢ (¬ ∀y ¬ ∀y ¬ ∀x∀yφ → ¬ ∀x∀yφ)
2	1	a3i 74	. 2 ⊢ (∀x∀yφ → ∀y ¬ ∀y ¬ ∀x∀yφ)
3		pm2.21 76	. . . . . 6 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → (∀x∀y ¬ ∀x∀yφ → ∀xφ))
4		ax467 1022	. . . . . 6 ⊢ ((∀x∀y ¬ ∀x∀yφ → ∀xφ) → φ)
5	3, 4	syl 10	. . . . 5 ⊢ (¬ ∀x∀y ¬ ∀x∀yφ → φ)
6	5	19.20i 991	. . . 4 ⊢ (∀x ¬ ∀x∀y ¬ ∀x∀yφ → ∀xφ)
7		ax467to6 1024	. . . 4 ⊢ (¬ ∀x ¬ ∀x∀y ¬ ∀x∀yφ → ∀y ¬ ∀x∀yφ)
8	6, 7	nsyl4 120	. . 3 ⊢ (¬ ∀y ¬ ∀x∀yφ → ∀xφ)
9	8	19.20i 991	. 2 ⊢ (∀y ¬ ∀y ¬ ∀x∀yφ → ∀y∀xφ)
10	2, 9	syl 10	1 ⊢ (∀x∀yφ → ∀y∀xφ)

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

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

Copyright terms: Public domain