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Mirrors > Home > ILE Home > Th. List > mpjao3dan | GIF version |
Description: Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
mpjao3dan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
mpjao3dan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
mpjao3dan.3 | ⊢ ((𝜑 ∧ 𝜏) → 𝜒) |
mpjao3dan.4 | ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏)) |
Ref | Expression |
---|---|
mpjao3dan | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpjao3dan.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | mpjao3dan.2 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
3 | 1, 2 | jaodan 710 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
4 | mpjao3dan.3 | . 2 ⊢ ((𝜑 ∧ 𝜏) → 𝜒) | |
5 | mpjao3dan.4 | . . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏)) | |
6 | df-3or 886 | . . 3 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜏)) | |
7 | 5, 6 | sylib 127 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ∨ 𝜏)) |
8 | 3, 4, 7 | mpjaodan 711 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∨ wo 629 ∨ w3o 884 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-3or 886 |
This theorem is referenced by: wetriext 4301 nntri3 6075 nntri2or2 6076 caucvgprlemnkj 6764 caucvgprlemnbj 6765 caucvgprprlemnkj 6790 caucvgprprlemnbj 6791 caucvgsr 6886 |
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