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Theorem infeq2 6763
 Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq2 (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

Proof of Theorem infeq2
StepHypRef Expression
1 supeq2 6738 . 2 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
2 df-inf 6734 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
3 df-inf 6734 . 2 inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, 𝑅)
41, 2, 33eqtr4g 2146 1 (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290  ◡ccnv 4450  supcsup 6731  infcinf 6732 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-uni 3660  df-sup 6733  df-inf 6734 This theorem is referenced by: (None)
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