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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 2253 | . 2 ⊢ (𝑅 = 𝑆 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 4019 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
3 | df-br 4019 | . 2 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
4 | 1, 2, 3 | 3bitr4g 223 | 1 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 〈cop 3610 class class class wbr 4018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-clel 2185 df-br 4019 |
This theorem is referenced by: breqi 4024 breqd 4029 poeq1 4314 soeq1 4330 frforeq1 4358 weeq1 4371 fveq1 5529 foeqcnvco 5807 f1eqcocnv 5808 isoeq2 5819 isoeq3 5820 ofreq 6104 supeq3 7007 tapeq1 7269 shftfvalg 10845 shftfval 10848 pw1nct 15150 |
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