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Mirrors > Home > MPE Home > Th. List > bitr | Structured version Visualization version GIF version |
Description: Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 274 in closed form. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
bitr | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bibi1 352 | . 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))) | |
2 | 1 | biimpar 478 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 |
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 206 df-an 397 |
This theorem is referenced by: opelopabt 5448 domunfican 9075 albitr 41963 3orbi123VD 42452 e2ebindALT 42531 |
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