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Theorem infeq2 9337
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 9306 . 2 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
2 df-inf 9301 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
3 df-inf 9301 . 2 inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, 𝑅)
41, 2, 33eqtr4g 2801 1 (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ccnv 5620  supcsup 9298  infcinf 9299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-uni 4854  df-sup 9300  df-inf 9301
This theorem is referenced by: (None)
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