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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 1144 | . . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜓) |
3 | syld3an2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
4 | 3 | 3com23 1144 | . . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜏) |
5 | 2, 4 | syld3an3 1214 | . 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜏) |
6 | 5 | 3com23 1144 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: nppcan2 7339 nnncan 7343 nnncan2 7345 ltdivmul 7954 ledivmul 7955 ltdiv23 7970 lediv23 7971 dvdssub2 10237 dvdsgcdb 10402 lcmdvdsb 10466 |
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