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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 399 | 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 504 ad4ant134 1195 r19.29an 2574 diffifi 6788 fimax2gtrilemstep 6794 cnegexlem3 7939 cnegex 7940 lemul12b 8619 climshftlemg 11071 prodeq2 11326 lcmdvds 11760 pw2dvdslemn 11843 tgcl 12233 metss 12663 ivthinclemlr 12784 ivthinclemur 12786 |
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