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Mirrors > Home > ILE Home > Th. List > eqbrtrdi | GIF version |
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) |
Ref | Expression |
---|---|
eqbrtrdi.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqbrtrdi.2 | ⊢ 𝐵𝑅𝐶 |
Ref | Expression |
---|---|
eqbrtrdi | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrtrdi.2 | . 2 ⊢ 𝐵𝑅𝐶 | |
2 | eqbrtrdi.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | breq1d 3997 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
4 | 1, 3 | mpbiri 167 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 class class class wbr 3987 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 |
This theorem is referenced by: eqbrtrrdi 4027 pm54.43 7160 nn0ledivnn 9717 xltnegi 9785 leexp1a 10524 facwordi 10667 faclbnd3 10670 resqrexlemlo 10970 efap0 11633 dvds1 11806 en1top 12836 dvef 13447 rpabscxpbnd 13618 zabsle1 13659 trirec0 14041 |
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