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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 3942 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)) |
5 | 1, 4 | mpbid 146 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 class class class wbr 3929 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 |
This theorem is referenced by: ofrval 5992 phplem2 6747 ltaddnq 7215 prarloclemarch2 7227 prmuloclemcalc 7373 axcaucvglemcau 7706 apreap 8349 ltmul1 8354 subap0d 8406 divap1d 8561 div2subap 8596 lemul2a 8617 mul2lt0rlt0 9546 xleadd2a 9657 monoord2 10250 expubnd 10350 bernneq2 10413 resqrexlemcalc2 10787 resqrexlemcalc3 10788 abs2dif2 10879 bdtrilem 11010 bdtri 11011 xrmaxaddlem 11029 fsum00 11231 iserabs 11244 geosergap 11275 mertenslemi1 11304 eftlub 11396 eirraplem 11483 xblss2 12574 xmstri2 12639 mstri2 12640 xmstri 12641 mstri 12642 xmstri3 12643 mstri3 12644 msrtri 12645 |
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