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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 2098 | . 2 ⊢ (𝑅 = 𝑆 → (〈A, B〉 ∈ 𝑅 ↔ 〈A, B〉 ∈ 𝑆)) | |
2 | df-br 3756 | . 2 ⊢ (A𝑅B ↔ 〈A, B〉 ∈ 𝑅) | |
3 | df-br 3756 | . 2 ⊢ (A𝑆B ↔ 〈A, B〉 ∈ 𝑆) | |
4 | 1, 2, 3 | 3bitr4g 212 | 1 ⊢ (𝑅 = 𝑆 → (A𝑅B ↔ A𝑆B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 〈cop 3370 class class class wbr 3755 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 df-clel 2033 df-br 3756 |
This theorem is referenced by: breqi 3761 breqd 3766 poeq1 4027 soeq1 4043 fveq1 5120 foeqcnvco 5373 f1eqcocnv 5374 isoeq2 5385 isoeq3 5386 ofreq 5657 |
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