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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc5c4c711toc5 | Structured version Visualization version GIF version |
Description: Rederivation of sp 2172 from axc5c4c711 40610. Note that ax6 2393 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1962 instead of ax6 2393, so that this rederivation requires only ax6v 1962 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc5c4c711toc5 | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6v 1962 | . . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | |
2 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))) | |
3 | ax-1 6 | . . . 4 ⊢ ((𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) → (∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦)))) | |
4 | axc5c4c711 40610 | . . . 4 ⊢ ((∀𝑥∀𝑥 ¬ ∀𝑥∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(∀𝑥𝜑 → ¬ 𝑥 = 𝑦))) → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦)) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (¬ 𝜑 → (∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦)) |
6 | 1, 5 | mtoi 200 | . 2 ⊢ (¬ 𝜑 → ¬ ∀𝑥𝜑) |
7 | 6 | con4i 114 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 = wceq 1528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: (None) |
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