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Mirrors > Home > ILE Home > Th. List > mp3anr3 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
Ref | Expression |
---|---|
mp3anr3.1 | ⊢ 𝜃 |
mp3anr3.2 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) |
Ref | Expression |
---|---|
mp3anr3 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3anr3.1 | . . 3 ⊢ 𝜃 | |
2 | mp3anr3.2 | . . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | |
3 | 2 | ancoms 266 | . . 3 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) → 𝜏) |
4 | 1, 3 | mp3anl3 1315 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜏) |
5 | 4 | ancoms 266 | 1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 965 |
This theorem is referenced by: rprelogbdiv 13316 |
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