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Mirrors > Home > ILE Home > Th. List > anandir | 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 394 | . . 3 ⊢ ((𝜒 ∧ 𝜒) ↔ 𝜒) | |
2 | 1 | anbi2i 453 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
3 | an4 576 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) | |
4 | 2, 3 | bitr3i 185 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: anandi3r 977 fununi 5237 imadiflem 5248 imadif 5249 imainlem 5250 elfzuzb 9917 |
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