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Mirrors > Home > ILE Home > Th. List > supeq1d | GIF version |
Description: Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
supeq1d.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
supeq1d | ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supeq1d.1 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | supeq1 6984 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 supcsup 6980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-uni 3810 df-sup 6982 |
This theorem is referenced by: sup3exmid 8913 supminfex 9596 minmax 11237 xrminmax 11272 xrminrecl 11280 xrminadd 11282 suprzubdc 11952 gcdval 11959 gcdass 12015 pceulem 12293 pceu 12294 pcval 12295 pczpre 12296 pcdiv 12301 pcneg 12323 prdsex 12717 xmetxp 13977 xmetxpbl 13978 txmetcnp 13988 qtopbasss 13991 |
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