![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mp3anl2 | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
mp3anl2.1 | ⊢ 𝜓 |
mp3anl2.2 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
mp3anl2 | ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3anl2.1 | . . 3 ⊢ 𝜓 | |
2 | mp3anl2.2 | . . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
3 | 2 | ex 414 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏)) |
4 | 1, 3 | mp3an2 1450 | . 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 → 𝜏)) |
5 | 4 | imp 408 | 1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 |
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 398 df-3an 1090 |
This theorem is referenced by: mp3anr2 1460 preleq 9606 1dvds 16209 bcs2 30412 nmopub2tALT 31139 nmfnleub2 31156 nmophmi 31261 nmopcoadji 31331 atordi 31614 mdsymlem5 31637 |
Copyright terms: Public domain | W3C validator |