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Mirrors > Home > ILE Home > Th. List > 3brtr3d | GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.) |
Ref | Expression |
---|---|
3brtr3d.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr3d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3brtr3d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3brtr3d | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr3d.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr3d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | 3brtr3d.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 2, 3 | breq12d 3912 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)) |
5 | 1, 4 | mpbid 146 | 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: ofrval 5960 phplem2 6715 ltaddnq 7183 prarloclemarch2 7195 prmuloclemcalc 7341 axcaucvglemcau 7674 apreap 8317 ltmul1 8322 subap0d 8374 divap1d 8529 div2subap 8564 lemul2a 8585 mul2lt0rlt0 9514 xleadd2a 9625 monoord2 10218 expubnd 10318 bernneq2 10381 resqrexlemcalc2 10755 resqrexlemcalc3 10756 abs2dif2 10847 bdtrilem 10978 bdtri 10979 xrmaxaddlem 10997 fsum00 11199 iserabs 11212 geosergap 11243 mertenslemi1 11272 eftlub 11323 eirraplem 11410 xblss2 12501 xmstri2 12566 mstri2 12567 xmstri 12568 mstri 12569 xmstri3 12570 mstri3 12571 msrtri 12572 |
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