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Mirrors > Home > ILE Home > Th. List > eqbrtrrd | GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
Ref | Expression |
---|---|
eqbrtrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqbrtrrd.2 | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Ref | Expression |
---|---|
eqbrtrrd | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eqcomd 2045 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqbrtrrd.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐶) | |
4 | 2, 3 | eqbrtrd 3784 | 1 ⊢ (𝜑 → 𝐵𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 class class class wbr 3764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 |
This theorem is referenced by: dftpos4 5878 phpm 6327 prmuloclemcalc 6663 mullocprlem 6668 cauappcvgprlemladdfl 6753 caucvgprlemopl 6767 caucvgprprlemloccalc 6782 caucvgprprlemopl 6795 ltadd1sr 6861 axarch 6965 lemulge11 7832 ltexp2a 9306 leexp2a 9307 nnlesq 9356 cvg1nlemcxze 9581 resqrexlemover 9608 resqrexlemlo 9611 resqrexlemnmsq 9615 resqrexlemnm 9616 leabs 9672 abs3dif 9701 abs2dif 9702 recn2 9837 imcn2 9838 iiserex 9859 nn0seqcvgd 9880 |
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