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Theorem infeq1 8623
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 8592 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
2 df-inf 8590 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
3 df-inf 8590 . 2 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
41, 2, 33eqtr4g 2857 1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  ccnv 5310  supcsup 8587  infcinf 8588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-uni 4628  df-sup 8589  df-inf 8590
This theorem is referenced by:  infeq1d  8624  infeq1i  8625  ramcl2lem  16043  odval  18263  submod  18294  ioorval  23679  uniioombllem6  23693  infleinf  40321  infxrpnf  40406
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