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Mirrors > Home > ILE Home > Th. List > breqi | GIF version |
Description: Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.) |
Ref | Expression |
---|---|
breqi.1 | ⊢ 𝑅 = 𝑆 |
Ref | Expression |
---|---|
breqi | ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqi.1 | . 2 ⊢ 𝑅 = 𝑆 | |
2 | breq 3926 | . 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 class class class wbr 3924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-cleq 2130 df-clel 2133 df-br 3925 |
This theorem is referenced by: f1ompt 5564 brtpos2 6141 tfrexlem 6224 brdifun 6449 ltpiord 7120 ltxrlt 7823 ltxr 9555 xmeterval 12593 |
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