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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc5c711to11 | Structured version Visualization version GIF version |
Description: Rederivation of ax-11 2158 from axc5c711 36669. Note that ax-c7 36636 and ax-11 2158 are not used by the rederivation. The use of alimi 1819 (which uses ax-c5 36634) is allowed since we have already proved axc5c711toc5 36670. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc5c711to11 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc5c711toc7 36671 | . . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ¬ ∀𝑥∀𝑦𝜑) | |
2 | 1 | con4i 114 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑) |
3 | pm2.21 123 | . . . . . 6 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → (∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)) | |
4 | axc5c711 36669 | . . . . . 6 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑) |
6 | 5 | alimi 1819 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) |
7 | axc5c711toc7 36671 | . . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑥∀𝑦𝜑) | |
8 | 6, 7 | nsyl4 161 | . . 3 ⊢ (¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) |
9 | 8 | alimi 1819 | . 2 ⊢ (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
10 | 2, 9 | syl 17 | 1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-11 2158 ax-c5 36634 ax-c4 36635 ax-c7 36636 |
This theorem is referenced by: (None) |
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