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Mirrors > Home > ILE Home > Th. List > breq | GIF version |
Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.) |
Ref | Expression |
---|---|
breq | ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2158 | . 2 ⊢ (𝑅 = 𝑆 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 3868 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
3 | df-br 3868 | . 2 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
4 | 1, 2, 3 | 3bitr4g 222 | 1 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∈ wcel 1445 〈cop 3469 class class class wbr 3867 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-4 1452 ax-17 1471 ax-ial 1479 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-cleq 2088 df-clel 2091 df-br 3868 |
This theorem is referenced by: breqi 3873 breqd 3878 poeq1 4150 soeq1 4166 frforeq1 4194 weeq1 4207 fveq1 5339 foeqcnvco 5607 f1eqcocnv 5608 isoeq2 5619 isoeq3 5620 ofreq 5897 supeq3 6765 shftfvalg 10383 shftfval 10386 |
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