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Mirrors > Home > ILE Home > Th. List > adantllr | GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) |
Ref | Expression |
---|---|
adantl2.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
adantllr | ⊢ ((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . 2 ⊢ ((𝜑 ∧ 𝜏) → 𝜑) | |
2 | adantl2.1 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | sylanl1 400 | 1 ⊢ ((((𝜑 ∧ 𝜏) ∧ 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: ad4ant13 510 ad4ant134 1212 r19.29an 2612 diffifi 6872 fimax2gtrilemstep 6878 cnegexlem3 8096 cnegex 8097 lemul12b 8777 climshftlemg 11265 prodeq2 11520 fprodmodd 11604 lcmdvds 12033 pw2dvdslemn 12119 tgcl 12858 metss 13288 ivthinclemlr 13409 ivthinclemur 13411 |
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