Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mp3anr1 | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
Ref | Expression |
---|---|
mp3anr1.1 | ⊢ 𝜓 |
mp3anr1.2 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) |
Ref | Expression |
---|---|
mp3anr1 | ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3anr1.1 | . . 3 ⊢ 𝜓 | |
2 | mp3anr1.2 | . . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | |
3 | 2 | ancoms 468 | . . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏) |
4 | 1, 3 | mp3anl1 1410 | . 2 ⊢ (((𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏) |
5 | 4 | ancoms 468 | 1 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 df-3an 1033 |
This theorem is referenced by: vc2OLD 26807 vc0 26813 vcm 26815 nvaddsub4 26896 nvpi 26906 nvge0 26912 ipval3 26948 ipidsq 26949 lnoadd 26997 lnosub 26998 dipsubdir 27087 |
Copyright terms: Public domain | W3C validator |