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Mirrors > Home > MPE Home > Th. List > 3jaodan | Structured version Visualization version GIF version |
Description: Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
3jaodan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3jaodan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
3jaodan.3 | ⊢ ((𝜑 ∧ 𝜏) → 𝜒) |
Ref | Expression |
---|---|
3jaodan | ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3jaodan.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 415 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 3jaodan.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
4 | 3 | ex 415 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) |
5 | 3jaodan.3 | . . . 4 ⊢ ((𝜑 ∧ 𝜏) → 𝜒) | |
6 | 5 | ex 415 | . . 3 ⊢ (𝜑 → (𝜏 → 𝜒)) |
7 | 2, 4, 6 | 3jaod 1424 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜏) → 𝜒)) |
8 | 7 | imp 409 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1082 |
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 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 |
This theorem is referenced by: mpjao3dan 1427 onzsl 7564 zeo 12071 xrltnsym 12533 xrlttri 12535 xrlttr 12536 qbtwnxr 12596 xltnegi 12612 xaddcom 12636 xnegdi 12644 xsubge0 12657 xrub 12708 bpoly3 15415 blssioo 23406 ismbf2d 24244 itg2seq 24346 eliccioo 30611 3ccased 32953 lineelsb2 33613 dfxlim2v 42134 |
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