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Mirrors > Home > MPE Home > Th. List > 4cases | Structured version Visualization version GIF version |
Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.) |
Ref | Expression |
---|---|
4cases.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
4cases.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) |
4cases.3 | ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒) |
4cases.4 | ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
4cases | ⊢ 𝜒 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4cases.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 4cases.3 | . . 3 ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒) | |
3 | 1, 2 | pm2.61ian 812 | . 2 ⊢ (𝜓 → 𝜒) |
4 | 4cases.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) | |
5 | 4cases.4 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒) | |
6 | 4, 5 | pm2.61ian 812 | . 2 ⊢ (¬ 𝜓 → 𝜒) |
7 | 3, 6 | pm2.61i 185 | 1 ⊢ 𝜒 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: 4casesdan 1041 suc11reg 9155 hasheqf1oi 13804 fvprmselgcd1 16481 axlowdimlem15 26902 ax12eq 36598 ax12el 36599 cdleme27a 38024 |
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