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Mirrors > Home > ILE Home > Th. List > 3adantl2 | GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
3adantl.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
3adantl2 | ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpb 995 | . 2 ⊢ ((𝜑 ∧ 𝜏 ∧ 𝜓) → (𝜑 ∧ 𝜓)) | |
2 | 3adantl.1 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | sylan 283 | 1 ⊢ (((𝜑 ∧ 𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 980 |
This theorem is referenced by: 3ad2antl1 1159 nnmord 6511 ltaprg 7596 lediv2a 8828 zdiv 9317 mulgnn0subcl 12872 mulgsubcl 12873 neiint 13278 cnpnei 13352 |
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