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Mirrors > Home > ILE Home > Th. List > 3bitrri | GIF version |
Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3bitri.1 | ⊢ (𝜑 ↔ 𝜓) |
3bitri.2 | ⊢ (𝜓 ↔ 𝜒) |
3bitri.3 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
3bitrri | ⊢ (𝜃 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitri.3 | . 2 ⊢ (𝜒 ↔ 𝜃) | |
2 | 3bitri.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 3bitri.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | bitr2i 184 | . 2 ⊢ (𝜒 ↔ 𝜑) |
5 | 1, 4 | bitr3i 185 | 1 ⊢ (𝜃 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: sbralie 2665 reu8 2875 unass 3228 ssin 3293 difab 3340 iunss 3849 poirr 4224 cnvuni 4720 dfco2 5033 dff1o6 5670 elznn0 9062 bj-ssom 13123 |
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