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Mirrors > Home > ILE Home > Th. List > adantlll | GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) |
Ref | Expression |
---|---|
adantl2.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
adantlll | ⊢ ((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . 2 ⊢ ((𝜏 ∧ 𝜑) → 𝜑) | |
2 | adantl2.1 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | sylanl1 400 | 1 ⊢ ((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
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 is referenced by: ad4ant23 507 ad4ant24 508 ad4ant234 1200 fun11iun 5435 fiintim 6873 cnegexlem3 8052 bezoutlemzz 11885 cnptopco 12622 |
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