Theorem ax67to7 1020

Description: Re-derivation of ax-7 960 from ax67 1018. Note that ax-6o 976 and ax-7 960 are not used by the re-derivation.

Assertion

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

Proof of Theorem ax67to7

Step	Hyp	Ref	Expression
1		ax67to6 1019	. . 3 ⊢ (¬ ∀y ¬ ∀y ¬ ∀x∀yφ → ¬ ∀x∀yφ)
2	1	a3i 74	. 2 ⊢ (∀x∀yφ → ∀y ¬ ∀y ¬ ∀x∀yφ)
3		ax67 1018	. . 3 ⊢ (¬ ∀y ¬ ∀x∀yφ → ∀xφ)
4	3	19.20i 990	. 2 ⊢ (∀y ¬ ∀y ¬ ∀x∀yφ → ∀y∀xφ)
5	2, 4	syl 10	1 ⊢ (∀x∀yφ → ∀y∀xφ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 952

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976

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