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Mirrors > Home > MPE Home > Th. List > mp2ani | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
mp2ani.1 | ⊢ 𝜓 |
mp2ani.2 | ⊢ 𝜒 |
mp2ani.3 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) |
Ref | Expression |
---|---|
mp2ani | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp2ani.2 | . 2 ⊢ 𝜒 | |
2 | mp2ani.1 | . . 3 ⊢ 𝜓 | |
3 | mp2ani.3 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | |
4 | 2, 3 | mpani 694 | . 2 ⊢ (𝜑 → (𝜒 → 𝜃)) |
5 | 1, 4 | mpi 20 | 1 ⊢ (𝜑 → 𝜃) |
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: dfom3 9110 dfac5lem4 9552 dfac9 9562 cflem 9668 canthp1lem2 10075 addsrpr 10497 mulsrpr 10498 trclublem 14355 gcdaddmlem 15872 tgjustf 26259 sto1i 30013 stji1i 30019 kur14lem9 32461 dfon2lem4 33031 rtrclex 39997 comptiunov2i 40071 |
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