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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 2142 | . 2 ⊢ (𝑅 = 𝑆 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 3786 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
3 | df-br 3786 | . 2 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
4 | 1, 2, 3 | 3bitr4g 221 | 1 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 df-br 3786 |
This theorem is referenced by: breqi 3791 breqd 3796 poeq1 4054 soeq1 4070 frforeq1 4098 weeq1 4111 fveq1 5197 foeqcnvco 5450 f1eqcocnv 5451 isoeq2 5462 isoeq3 5463 ofreq 5735 supeq3 6403 shftfvalg 9706 shftfval 9709 |
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