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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 4003 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 class class class wbr 4000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 |
This theorem is referenced by: eqbrtri 4021 brtpos0 6246 euen1 6795 euen1b 6796 2dom 6798 infglbti 7017 pr2nelem 7183 caucvgprprlemnbj 7670 caucvgprprlemmu 7672 caucvgprprlemaddq 7685 caucvgprprlem1 7686 gt0srpr 7725 caucvgsr 7779 mappsrprg 7781 map2psrprg 7782 pitonnlem1 7822 pitoregt0 7826 axprecex 7857 axpre-mulgt0 7864 axcaucvglemres 7876 lt0neg1 8402 le0neg1 8404 reclt1 8829 addltmul 9131 eluz2b1 9577 nn01to3 9593 xlt0neg1 9812 xle0neg1 9814 iccshftr 9968 iccshftl 9970 iccdil 9972 icccntr 9974 bernneq 10613 cbvsum 11339 expcnv 11483 cbvprod 11537 oddge22np1 11856 nn0o1gt2 11880 isprm3 12088 dvdsnprmd 12095 pw2dvdslemn 12135 txmetcnp 13651 sincosq1sgn 13880 sincosq3sgn 13882 sincosq4sgn 13883 logrpap0b 13930 |
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