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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 4034 | . 2 ⊢ (𝜑 → (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)) |
5 | 1, 4 | mpbird 167 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 class class class wbr 4021 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 |
This theorem is referenced by: f1oiso2 5852 prarloclemarch2 7453 caucvgprprlemmu 7729 caucvgsrlembound 7828 mulap0 8646 lediv12a 8886 recp1lt1 8891 xleadd1a 9909 fldiv4p1lem1div2 10342 intfracq 10357 modqmulnn 10379 addmodlteq 10435 frecfzennn 10463 monoord2 10516 expgt1 10598 leexp2r 10614 leexp1a 10615 bernneq 10681 faclbnd 10762 faclbnd6 10765 facubnd 10766 hashunlem 10825 zfz1isolemiso 10860 sqrtgt0 11084 absrele 11133 absimle 11134 abstri 11154 abs2difabs 11158 bdtrilem 11288 bdtri 11289 xrmaxifle 11295 xrmaxadd 11310 xrbdtri 11325 climsqz 11384 climsqz2 11385 fsum3cvg2 11443 isumle 11544 expcnvap0 11551 expcnvre 11552 explecnv 11554 cvgratz 11581 efcllemp 11707 ege2le3 11720 eflegeo 11750 cos12dec 11816 phibnd 12260 pcdvdstr 12370 pcprmpw2 12376 pockthg 12400 psmetres2 14318 xmetres2 14364 comet 14484 bdxmet 14486 cnmet 14515 ivthdec 14607 limcimolemlt 14618 tangtx 14744 logbgcd1irraplemap 14872 cvgcmp2nlemabs 15268 trilpolemlt1 15277 |
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