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Mirrors > Home > NFE Home > Th. List > ax67to7 | GIF version |
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.) |
Ref | Expression |
---|---|
ax67to7 | ⊢ (∀x∀yφ → ∀y∀xφ) |
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) |
Colors of variables: wff setvar class |
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) |
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