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Mirrors > Home > ILE Home > Th. List > supeq1i | GIF version |
Description: Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
supeq1i.1 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
supeq1i | ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supeq1i.1 | . 2 ⊢ 𝐵 = 𝐶 | |
2 | supeq1 6681 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅) |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 supcsup 6677 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-uni 3654 df-sup 6679 |
This theorem is referenced by: infrenegsupex 9082 maxcom 10636 gcdcom 11243 gcdass 11282 |
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