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Mirrors > Home > ILE Home > Th. List > biantr | GIF version |
Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
biantr | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓)) | |
2 | 1 | bibi2d 221 | . 2 ⊢ ((𝜒 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜓))) |
3 | 2 | biimparc 283 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
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 |
This theorem is referenced by: bm1.1 2025 |
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