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Mirrors > Home > MPE Home > Th. List > imdistanri | Structured version Visualization version GIF version |
Description: Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
imdistanri.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
imdistanri | ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imdistanri.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | com12 32 | . 2 ⊢ (𝜓 → (𝜑 → 𝜒)) |
3 | 2 | impac 555 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: tc2 9178 prmodvdslcmf 16377 monmat2matmon 21426 cnextcn 22669 umgredg 26917 crctcshwlkn0lem5 27586 tpr2rico 31150 bj-snsetex 34270 bj-restuni 34382 poimirlem26 34912 seqpo 35016 isdrngo2 35230 pm10.55 40694 2pm13.193VD 41230 |
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