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Mirrors > Home > ILE Home > Th. List > 3brtr4d | GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
3brtr4d.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3brtr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3brtr4d | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr4d.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr4d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
3 | 3brtr4d.3 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) | |
4 | 2, 3 | breq12d 3912 | . 2 ⊢ (𝜑 → (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)) |
5 | 1, 4 | mpbird 166 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: f1oiso2 5696 prarloclemarch2 7195 caucvgprprlemmu 7471 caucvgsrlembound 7570 mulap0 8382 lediv12a 8616 recp1lt1 8621 xleadd1a 9611 fldiv4p1lem1div2 10033 intfracq 10048 modqmulnn 10070 addmodlteq 10126 frecfzennn 10154 monoord2 10205 expgt1 10286 leexp2r 10302 leexp1a 10303 bernneq 10367 faclbnd 10442 faclbnd6 10445 facubnd 10446 hashunlem 10505 zfz1isolemiso 10537 sqrtgt0 10761 absrele 10810 absimle 10811 abstri 10831 abs2difabs 10835 bdtrilem 10965 bdtri 10966 xrmaxifle 10970 xrmaxadd 10985 xrbdtri 11000 climsqz 11059 climsqz2 11060 fsum3cvg2 11118 isumle 11219 expcnvap0 11226 expcnvre 11227 explecnv 11229 cvgratz 11256 efcllemp 11278 ege2le3 11291 eflegeo 11322 cos12dec 11388 phibnd 11804 psmetres2 12413 xmetres2 12459 comet 12579 bdxmet 12581 cnmet 12610 ivthdec 12702 limcimolemlt 12713 cvgcmp2nlemabs 13123 trilpolemlt1 13130 |
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