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Mirrors > Home > MPE Home > Th. List > mp3anl1 | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
mp3anl1.1 | ⊢ 𝜑 |
mp3anl1.2 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
mp3anl1 | ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3anl1.1 | . . 3 ⊢ 𝜑 | |
2 | mp3anl1.2 | . . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
3 | 2 | ex 416 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏)) |
4 | 1, 3 | mp3an1 1449 | . 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜃 → 𝜏)) |
5 | 4 | imp 410 | 1 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ 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 210 df-an 400 df-3an 1090 |
This theorem is referenced by: mp3anr1 1459 facavg 13765 iddvds 15727 isprm7 16161 blometi 28750 mdslmd3i 30279 atcvat2i 30334 chirredlem3 30339 mdsymlem1 30350 |
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