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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc5c711 | Structured version Visualization version GIF version |
Description: Proof of a single axiom that can replace ax-c5 36583, ax-c7 36585, and ax-11 2160 in a subsystem that includes these axioms plus ax-c4 36584 and ax-gen 1803 (and propositional calculus). See axc5c711toc5 36619, axc5c711toc7 36620, and axc5c711to11 36621 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 36611. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc5c711 | ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c5 36583 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
2 | ax10fromc7 36595 | . . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑) | |
3 | ax-c7 36585 | . . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑) | |
4 | 3 | con1i 149 | . . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑) |
5 | 4 | alimi 1819 | . . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑) |
6 | ax-11 2160 | . . . 4 ⊢ (∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑) | |
7 | 2, 5, 6 | 3syl 18 | . . 3 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑) |
8 | 1, 7 | nsyl4 161 | . 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑) |
9 | ax-c5 36583 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
10 | 8, 9 | ja 189 | 1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀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 2160 ax-c5 36583 ax-c4 36584 ax-c7 36585 |
This theorem is referenced by: axc5c711toc5 36619 axc5c711toc7 36620 axc5c711to11 36621 |
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