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Theorem infeq2 8540
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 8509 . 2 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
2 df-inf 8504 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
3 df-inf 8504 . 2 inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, 𝑅)
41, 2, 33eqtr4g 2829 1 (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 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: (None)
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