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Mirrors > Home > ILE Home > Th. List > syl3anl1 | 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 1180 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
3 | syl3anl1.2 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
4 | 2, 3 | sylan 281 | 1 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 |
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 975 |
This theorem is referenced by: xrmaxaddlem 11210 lgsdinn0 13664 |
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