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Mirrors > Home > MPE Home > Th. List > imdistand | Structured version Visualization version GIF version |
Description: Distribution of implication with conjunction (deduction rule). (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 721 | . 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) ↔ ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | |
3 | 1, 2 | sylib 207 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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: imdistanda 725 a2and 849 predpo 5615 unblem1 8097 cfub 8954 lbzbi 11652 poimirlem32 32611 ispridl2 33007 ispridlc 33039 lnr2i 36705 rfovcnvf1od 37318 |
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