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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 6881 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 supcsup 6877 |
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-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-uni 3745 df-sup 6879 |
This theorem is referenced by: infrenegsupex 9416 maxcom 11007 xrmax2sup 11055 xrmaxltsup 11059 xrmaxadd 11062 infxrnegsupex 11064 gcdcom 11698 gcdass 11739 |
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