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Mirrors > Home > MPE Home > Th. List > Mathboxes > exmid2 | Structured version Visualization version GIF version |
Description: An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.) |
Ref | Expression |
---|---|
exmid2.1 | ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
exmid2.2 | ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒) |
Ref | Expression |
---|---|
exmid2 | ⊢ ((𝜑 ∧ 𝜂) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝜂) → 𝜑) | |
2 | 1 | anim2i 616 | . . . 4 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜂)) → (𝜓 ∧ 𝜑)) |
3 | 2 | ancoms 458 | . . 3 ⊢ (((𝜑 ∧ 𝜂) ∧ 𝜓) → (𝜓 ∧ 𝜑)) |
4 | exmid2.1 | . . 3 ⊢ ((𝜓 ∧ 𝜑) → 𝜒) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (((𝜑 ∧ 𝜂) ∧ 𝜓) → 𝜒) |
6 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝜂) → 𝜂) | |
7 | 6 | anim2i 616 | . . . 4 ⊢ ((¬ 𝜓 ∧ (𝜑 ∧ 𝜂)) → (¬ 𝜓 ∧ 𝜂)) |
8 | 7 | ancoms 458 | . . 3 ⊢ (((𝜑 ∧ 𝜂) ∧ ¬ 𝜓) → (¬ 𝜓 ∧ 𝜂)) |
9 | exmid2.2 | . . 3 ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (((𝜑 ∧ 𝜂) ∧ ¬ 𝜓) → 𝜒) |
11 | 5, 10 | pm2.61dan 809 | 1 ⊢ ((𝜑 ∧ 𝜂) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 396 |
This theorem is referenced by: (None) |
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