![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imdistand | Structured version Visualization version GIF version |
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 568 | . 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) ↔ ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | |
3 | 1, 2 | sylib 217 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
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 206 df-an 397 |
This theorem is referenced by: imdistanda 572 a2and 843 reximdvai 3165 unblem1 9297 cfub 10246 lbzbi 12922 sltlpss 27409 cusgredgex 34181 poimirlem32 36606 ispridl2 36992 ispridlc 37024 lnr2i 41940 rfovcnvf1od 42837 |
Copyright terms: Public domain | W3C validator |