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Mirrors > Home > ILE Home > Th. List > breq2i | GIF version |
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
Ref | Expression |
---|---|
breq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
breq2i | ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | breq2 3789 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1284 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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 |
This theorem is referenced by: breqtri 3808 en1 6302 snnen2og 6345 1nen2 6347 pm54.43 6459 caucvgprprlemval 6878 caucvgprprlemmu 6885 caucvgsr 6978 pitonnlem1 7013 lt0neg2 7573 le0neg2 7575 negap0 7729 recexaplem2 7742 recgt1 7975 crap0 8035 addltmul 8267 nn0lt10b 8428 nn0lt2 8429 3halfnz 8444 xlt0neg2 8906 xle0neg2 8908 iccshftr 9016 iccshftl 9018 iccdil 9020 icccntr 9022 cjap0 9794 abs00ap 9948 3dvdsdec 10264 3dvds2dec 10265 ndvdsi 10333 3prm 10510 prmfac1 10531 |
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