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Mirrors > Home > ILE Home > Th. List > jaoian | GIF version |
Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
Ref | Expression |
---|---|
jaoian.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
jaoian.2 | ⊢ ((𝜃 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
jaoian | ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaoian.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 114 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | jaoian.2 | . . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒) | |
4 | 3 | ex 114 | . . 3 ⊢ (𝜃 → (𝜓 → 𝜒)) |
5 | 2, 4 | jaoi 711 | . 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒)) |
6 | 5 | imp 123 | 1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ordi 811 ccase 959 xaddnemnf 9803 xaddnepnf 9804 faclbnd 10664 faclbnd3 10666 |
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