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Mirrors > Home > ILE Home > Th. List > andi | GIF version |
Description: Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) |
Ref | Expression |
---|---|
andi | ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 712 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) | |
2 | olc 711 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) | |
3 | 1, 2 | jaodan 797 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) |
4 | orc 712 | . . . 4 ⊢ (𝜓 → (𝜓 ∨ 𝜒)) | |
5 | 4 | anim2i 342 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ (𝜓 ∨ 𝜒))) |
6 | olc 711 | . . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒)) | |
7 | 6 | anim2i 342 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∧ (𝜓 ∨ 𝜒))) |
8 | 5, 7 | jaoi 716 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒))) |
9 | 3, 8 | impbii 126 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: andir 819 anddi 821 dcim 841 dcan 933 excxor 1378 sbequilem 1838 sborv 1890 r19.43 2635 indi 3384 difindiss 3391 unrab 3408 unipr 3825 uniun 3830 unopab 4084 xpundi 4684 coundir 5133 unpreima 5643 tpostpos 6267 elni2 7315 elznn0nn 9269 |
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