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Mirrors > Home > ILE Home > Th. List > syl3anl3 | GIF version |
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
syl3anl3.1 | ⊢ (𝜑 → 𝜃) |
syl3anl3.2 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl3anl3 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anl3.1 | . . 3 ⊢ (𝜑 → 𝜃) | |
2 | 1 | 3anim3i 1126 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
3 | syl3anl3.2 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
4 | 2, 3 | sylan 277 | 1 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: (None) |
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