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Mirrors > Home > MPE Home > Th. List > syl3anb | Structured version Visualization version GIF version |
Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
syl3anb.1 | ⊢ (𝜑 ↔ 𝜓) |
syl3anb.2 | ⊢ (𝜒 ↔ 𝜃) |
syl3anb.3 | ⊢ (𝜏 ↔ 𝜂) |
syl3anb.4 | ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) |
Ref | Expression |
---|---|
syl3anb | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anb.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | syl3anb.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
3 | syl3anb.3 | . . 3 ⊢ (𝜏 ↔ 𝜂) | |
4 | 1, 2, 3 | 3anbi123i 1135 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂)) |
5 | syl3anb.4 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) | |
6 | 4, 5 | sylbi 209 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 199 df-an 388 df-3an 1070 |
This theorem is referenced by: syl3anbr 1142 poxp 7620 infempty 8758 symgsssg 18346 symgfisg 18347 lmodvscl 19363 xrs1mnd 20275 iscnp2 21541 clwwlknccat 27577 slmdvscl 30464 cgr3permute3 32969 cgr3permute1 32970 cgr3permute2 32971 cgr3permute4 32972 cgr3permute5 32973 colinearxfr 32997 grposnOLD 34550 rngunsnply 39114 |
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