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Mirrors > Home > MPE Home > Th. List > anandir | Structured version Visualization version GIF version |
Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
anandir | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anidm 674 | . . 3 ⊢ ((𝜒 ∧ 𝜒) ↔ 𝜒) | |
2 | 1 | anbi2i 726 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
3 | an4 861 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) | |
4 | 2, 3 | bitr3i 265 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: anandi3r 1046 disjxun 4581 fununi 5878 imadif 5887 wfrlem5 7306 elfzuzb 12207 frgra3v 26529 5oalem3 27899 5oalem5 27901 frrlem5 31028 nzin 37539 un2122 38038 frgr3v 41445 |
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