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Mirrors > Home > ILE Home > Th. List > breq1i | GIF version |
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
Ref | Expression |
---|---|
breq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
breq1i | ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | breq1 3767 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
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: eqbrtri 3783 brtpos0 5867 euen1 6282 euen1b 6283 2dom 6285 caucvgprprlemnbj 6791 caucvgprprlemmu 6793 caucvgprprlemaddq 6806 caucvgprprlem1 6807 gt0srpr 6833 caucvgsr 6886 pitonnlem1 6921 pitoregt0 6925 axprecex 6954 axpre-mulgt0 6961 axcaucvglemres 6973 lt0neg1 7463 le0neg1 7465 reclt1 7862 addltmul 8161 eluz2b1 8539 nn01to3 8552 xlt0neg1 8751 xle0neg1 8753 iccshftr 8862 iccshftl 8864 iccdil 8866 icccntr 8868 bernneq 9369 |
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