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Theorem infeq1 8537
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 8506 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
2 df-inf 8504 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
3 df-inf 8504 . 2 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
41, 2, 33eqtr4g 2829 1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  ccnv 5248  supcsup 8501  infcinf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-uni 4573  df-sup 8503  df-inf 8504
This theorem is referenced by:  infeq1d  8538  infeq1i  8539  ramcl2lem  15919  odval  18159  submod  18190  ioorval  23561  uniioombllem6  23575  infleinf  40098  infxrpnf  40184
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