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Theorem infeq1 8379
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Proof of Theorem infeq1
StepHypRef Expression
1 supeq1 8348 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
2 df-inf 8346 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
3 df-inf 8346 . 2 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
41, 2, 33eqtr4g 2680 1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  ccnv 5111  supcsup 8343  infcinf 8344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-uni 4435  df-sup 8345  df-inf 8346
This theorem is referenced by:  infeq1d  8380  infeq1i  8381  ramcl2lem  15707  odval  17947  submod  17978  ioorval  23336  uniioombllem6  23350  infleinf  39407  infxrpnf  39493
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