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 6963 | . 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 1348 supcsup 6959 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-uni 3797 df-sup 6961 |
This theorem is referenced by: sup3exmid 8873 supminfex 9556 minmax 11193 xrminmax 11228 xrminrecl 11236 xrminadd 11238 suprzubdc 11907 gcdval 11914 gcdass 11970 pceulem 12248 pceu 12249 pcval 12250 pczpre 12251 pcdiv 12256 pcneg 12278 xmetxp 13301 xmetxpbl 13302 txmetcnp 13312 qtopbasss 13315 |
Copyright terms: Public domain | W3C validator |