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Mirrors > Home > ILE Home > Th. List > pm4.78i | GIF version |
Description: Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.) |
Ref | Expression |
---|---|
pm4.78i | ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 633 | . . 3 ⊢ (𝜓 → (𝜓 ∨ 𝜒)) | |
2 | 1 | imim2i 12 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∨ 𝜒))) |
3 | olc 632 | . . 3 ⊢ (𝜒 → (𝜓 ∨ 𝜒)) | |
4 | 3 | imim2i 12 | . 2 ⊢ ((𝜑 → 𝜒) → (𝜑 → (𝜓 ∨ 𝜒))) |
5 | 2, 4 | jaoi 636 | 1 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 |
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 |
This theorem is referenced by: (None) |
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