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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 3902 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 class class class wbr 3899 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 |
This theorem is referenced by: eqbrtri 3919 brtpos0 6117 euen1 6664 euen1b 6665 2dom 6667 infglbti 6880 pr2nelem 7015 caucvgprprlemnbj 7469 caucvgprprlemmu 7471 caucvgprprlemaddq 7484 caucvgprprlem1 7485 gt0srpr 7524 caucvgsr 7578 mappsrprg 7580 map2psrprg 7581 pitonnlem1 7621 pitoregt0 7625 axprecex 7656 axpre-mulgt0 7663 axcaucvglemres 7675 lt0neg1 8198 le0neg1 8200 reclt1 8622 addltmul 8924 eluz2b1 9363 nn01to3 9377 xlt0neg1 9589 xle0neg1 9591 iccshftr 9745 iccshftl 9747 iccdil 9749 icccntr 9751 bernneq 10380 cbvsum 11097 expcnv 11241 oddge22np1 11505 nn0o1gt2 11529 isprm3 11726 dvdsnprmd 11733 pw2dvdslemn 11770 txmetcnp 12614 sincosq1sgn 12834 sincosq3sgn 12836 sincosq4sgn 12837 |
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