Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bitr | Structured version Visualization version GIF version |
Description: Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 276 in closed form. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
bitr | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bibi1 353 | . 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 207 ∧ 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 208 df-an 397 |
This theorem is referenced by: opelopabt 5410 domunfican 8779 albitr 40572 3orbi123VD 41061 e2ebindALT 41140 |
Copyright terms: Public domain | W3C validator |