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Mirrors > Home > ILE Home > Th. List > syld3an2 | GIF version |
Description: A syllogism inference. (Contributed by NM, 20-May-2007.) |
Ref | Expression |
---|---|
syld3an2.1 | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓) |
syld3an2.2 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syld3an2 | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syld3an2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓) | |
2 | 1 | 3com23 1188 | . . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜓) |
3 | syld3an2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
4 | 3 | 3com23 1188 | . . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜏) |
5 | 2, 4 | syld3an3 1262 | . 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜏) |
6 | 5 | 3com23 1188 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 |
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 965 |
This theorem is referenced by: nppcan2 8017 nnncan 8021 nnncan2 8023 ltdivmul 8658 ledivmul 8659 ltdiv23 8674 lediv23 8675 dvdssub2 11571 dvdsgcdb 11737 lcmdvdsb 11801 rpdivcxp 13040 |
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