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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc711toc7 | Structured version Visualization version GIF version |
Description: Rederivation of ax-c7 38351 from axc711 38380. Note that ax-c7 38351 and ax-11 2147 are not used by the rederivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc711toc7 | ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1-o 38363 | . . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
2 | 1 | con3i 154 | . . . 4 ⊢ (¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) |
3 | 2 | alimi 1806 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
4 | 3 | con3i 154 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑) |
5 | axc711 38380 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
6 | ax-c5 38349 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
7 | 4, 5, 6 | 3syl 18 | 1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-11 2147 ax-c5 38349 ax-c4 38350 ax-c7 38351 |
This theorem is referenced by: axc711to11 38383 |
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