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| Description: Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.) | 
| Ref | Expression | 
|---|---|
| imdistand.1 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | 
| Ref | Expression | 
|---|---|
| imdistand | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | imdistand.1 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | |
| 2 | imdistan 567 | . 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) ↔ ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 | 
| 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 207 df-an 396 | 
| This theorem is referenced by: imdistanda 571 a2and 845 reximdvai 3164 unblem1 9329 cfub 10290 lbzbi 12979 sltlpss 27946 cusgredgex 35128 poimirlem32 37660 ispridl2 38046 ispridlc 38078 lnr2i 43133 rfovcnvf1od 44022 | 
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