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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 3768 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 class class class wbr 3764 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 |
This theorem is referenced by: breqtri 3787 en1 6279 snnen2og 6322 caucvgprprlemval 6786 caucvgprprlemmu 6793 caucvgsr 6886 pitonnlem1 6921 lt0neg2 7464 le0neg2 7466 negap0 7620 recexaplem2 7633 recgt1 7863 crap0 7910 addltmul 8161 nn0lt10b 8321 nn0lt2 8322 xlt0neg2 8752 xle0neg2 8754 iccshftr 8862 iccshftl 8864 iccdil 8866 icccntr 8868 cjap0 9507 abs00ap 9660 |
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