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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 35493, ax-c7 35495, and ax-11 2093 in a subsystem that includes these axioms plus ax-c4 35494 and ax-gen 1758 (and propositional calculus). See axc5c711toc5 35529, axc5c711toc7 35530, and axc5c711to11 35531 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 35521. (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 35493 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
2 | ax10fromc7 35505 | . . . 4 ⊢ (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑) | |
3 | ax-c7 35495 | . . . . . 6 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑦𝜑) | |
4 | 3 | con1i 147 | . . . . 5 ⊢ (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥∀𝑦𝜑) |
5 | 4 | alimi 1774 | . . . 4 ⊢ (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦∀𝑥 ¬ ∀𝑥∀𝑦𝜑) |
6 | ax-11 2093 | . . . 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 158 | . 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → 𝜑) |
9 | ax-c5 35493 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
10 | 8, 9 | ja 175 | 1 ⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦𝜑 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-11 2093 ax-c5 35493 ax-c4 35494 ax-c7 35495 |
This theorem is referenced by: axc5c711toc5 35529 axc5c711toc7 35530 axc5c711to11 35531 |
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