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Mirrors > Home > MPE Home > Th. List > syl3anl1 | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
syl3anl1.1 | ⊢ (𝜑 → 𝜓) |
syl3anl1.2 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl3anl1 | ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anl1.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | 3anim1i 1151 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
3 | syl3anl1.2 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
4 | 2, 3 | sylan 580 | 1 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 |
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 397 df-3an 1088 |
This theorem is referenced by: suprzcl 12400 latjcom 18163 latmcom 18179 ring1zr 20544 lgsdinn0 26491 revpfxsfxrev 33073 crngohomfo 36160 dalem53 37735 |
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