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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 3950 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)) |
5 | 1, 4 | mpbid 146 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 class class class wbr 3937 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 |
This theorem is referenced by: ofrval 6000 phplem2 6755 ltaddnq 7239 prarloclemarch2 7251 prmuloclemcalc 7397 axcaucvglemcau 7730 apreap 8373 ltmul1 8378 subap0d 8430 divap1d 8585 div2subap 8620 lemul2a 8641 mul2lt0rlt0 9576 xleadd2a 9687 monoord2 10281 expubnd 10381 bernneq2 10444 apexp1 10496 resqrexlemcalc2 10819 resqrexlemcalc3 10820 abs2dif2 10911 bdtrilem 11042 bdtri 11043 xrmaxaddlem 11061 fsum00 11263 iserabs 11276 geosergap 11307 mertenslemi1 11336 eftlub 11433 eirraplem 11519 xblss2 12613 xmstri2 12678 mstri2 12679 xmstri 12680 mstri 12681 xmstri3 12682 mstri3 12683 msrtri 12684 logdivlti 13010 apdifflemr 13415 |
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