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Mirrors > Home > ILE Home > Th. List > andir | GIF version |
Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
andir | ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andi 819 | . 2 ⊢ ((𝜒 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓))) | |
2 | ancom 266 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑 ∨ 𝜓))) | |
3 | ancom 266 | . . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜑)) | |
4 | ancom 266 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)) | |
5 | 3, 4 | orbi12i 765 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓))) |
6 | 1, 2, 5 | 3bitr4i 212 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 709 |
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 710 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: anddi 822 excxor 1389 xordc1 1404 sbequilem 1849 rexun 3330 rabun2 3429 reuun2 3433 xpundir 4698 coundi 5145 mptun 5363 tpostpos 6284 ltxr 9800 pythagtriplem2 12293 pythagtrip 12310 |
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