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Mirrors > Home > ILE Home > Th. List > syld3an3 | GIF version |
Description: A syllogism inference. (Contributed by NM, 20-May-2007.) |
Ref | Expression |
---|---|
syld3an3.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
syld3an3.2 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syld3an3 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑) | |
2 | simp2 982 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | |
3 | syld3an3.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
4 | syld3an3.2 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
5 | 1, 2, 3, 4 | syl3anc 1216 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 |
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 depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: syld3an1 1262 syld3an2 1263 brelrng 4770 moriotass 5758 nnncan1 7998 lediv1 8627 modqval 10097 modqvalr 10098 modqcl 10099 flqpmodeq 10100 modq0 10102 modqge0 10105 modqlt 10106 modqdiffl 10108 modqdifz 10109 modqvalp1 10116 exp3val 10295 bcval4 10498 dvdsmultr1 11531 dvdssub2 11535 divalglemeuneg 11620 ndvdsadd 11628 basgen2 12250 opnneiss 12327 cnpf2 12376 sincosq1lem 12906 |
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