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Theorem infeq3 9504
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 5876 . . 3 (𝑅 = 𝑆𝑅 = 𝑆)
2 supeq3 9473 . . 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 9467 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
5 df-inf 9467 . 2 inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, 𝑆)
63, 4, 53eqtr4g 2793 1 (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  ccnv 5677  supcsup 9464  infcinf 9465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5686  df-sup 9466  df-inf 9467
This theorem is referenced by: (None)
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