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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 2042 | . 2 ⊢ (φ → B = A) |
3 | eqbrtrrd.2 | . 2 ⊢ (φ → A𝑅𝐶) | |
4 | 2, 3 | eqbrtrd 3775 | 1 ⊢ (φ → B𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 class class class wbr 3755 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 |
This theorem is referenced by: dftpos4 5819 prmuloclemcalc 6546 mullocprlem 6551 cauappcvgprlemladdfl 6627 caucvgprlemopl 6640 axarch 6773 lemulge11 7613 ltexp2a 8960 leexp2a 8961 nnlesq 9009 |
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