Theorem ax67 1019

Description: Proof of a single axiom that can replace both ax-6o 977 and ax-7 961. See ax67to6 1020 and ax67to7 1021 for the re-derivation of those axioms.

Assertion

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

Proof of Theorem ax67

Step	Hyp	Ref	Expression
1		ax-7 961	. . . . 5 ⊢ (∀y∀xφ → ∀x∀yφ)
2	1	con3i 98	. . . 4 ⊢ (¬ ∀x∀yφ → ¬ ∀y∀xφ)
3	2	19.20i 991	. . 3 ⊢ (∀x ¬ ∀x∀yφ → ∀x ¬ ∀y∀xφ)
4	3	con3i 98	. 2 ⊢ (¬ ∀x ¬ ∀y∀xφ → ¬ ∀x ¬ ∀x∀yφ)
5		ax-6o 977	. 2 ⊢ (¬ ∀x ¬ ∀x∀yφ → ∀yφ)
6	4, 5	syl 10	1 ⊢ (¬ ∀x ¬ ∀y∀xφ → ∀yφ)

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

This theorem is referenced by:  ax67to6 1020  ax67to7 1021

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

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