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Mirrors > Home > ILE Home > Th. List > syl3an | GIF version |
Description: A triple syllogism inference. (Contributed by NM, 13-May-2004.) |
Ref | Expression |
---|---|
syl3an.1 | ⊢ (𝜑 → 𝜓) |
syl3an.2 | ⊢ (𝜒 → 𝜃) |
syl3an.3 | ⊢ (𝜏 → 𝜂) |
syl3an.4 | ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) |
Ref | Expression |
---|---|
syl3an | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | syl3an.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | syl3an.3 | . . 3 ⊢ (𝜏 → 𝜂) | |
4 | 1, 2, 3 | 3anim123i 1089 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) |
5 | syl3an.4 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) | |
6 | 4, 5 | syl 14 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 885 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-3an 887 |
This theorem is referenced by: funtpg 4950 ftpg 5347 eloprabga 5591 addasspig 6428 mulasspig 6430 distrpig 6431 addcanpig 6432 mulcanpig 6433 ltapig 6436 distrnqg 6485 distrnq0 6557 cnegexlem2 7187 zletr 8294 zdivadd 8329 iooneg 8856 fzen 8907 fzaddel 8922 fzrev 8946 fzrevral2 8968 fzshftral 8970 fzosubel2 9051 fzonn0p1p1 9069 resqrexlemover 9608 |
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