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Theorem infeq3 9520
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq3 (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Proof of Theorem infeq3
StepHypRef Expression
1 cnveq 5884 . . 3 (𝑅 = 𝑆𝑅 = 𝑆)
2 supeq3 9489 . . 3 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
31, 2syl 17 . 2 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
4 df-inf 9483 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
5 df-inf 9483 . 2 inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, 𝑆)
63, 4, 53eqtr4g 2802 1 (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ccnv 5684  supcsup 9480  infcinf 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-sup 9482  df-inf 9483
This theorem is referenced by: (None)
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