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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 6985 | . 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 6981 |
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 3811 df-sup 6983 |
This theorem is referenced by: sup3exmid 8914 supminfex 9597 minmax 11238 xrminmax 11273 xrminrecl 11281 xrminadd 11283 suprzubdc 11953 gcdval 11960 gcdass 12016 pceulem 12294 pceu 12295 pcval 12296 pczpre 12297 pcdiv 12302 pcneg 12324 prdsex 12718 xmetxp 14010 xmetxpbl 14011 txmetcnp 14021 qtopbasss 14024 |
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