Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6931 | . 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 1335 supcsup 6927 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-uni 3774 df-sup 6929 |
This theorem is referenced by: sup3exmid 8829 supminfex 9509 minmax 11133 xrminmax 11166 xrminrecl 11174 xrminadd 11176 gcdval 11847 gcdass 11903 xmetxp 12949 xmetxpbl 12950 txmetcnp 12960 qtopbasss 12963 |
Copyright terms: Public domain | W3C validator |