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Theorem infeq3 8269
 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 5218 . . 3 (𝑅 = 𝑆𝑅 = 𝑆)
2 supeq3 8238 . . 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 8232 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
5 df-inf 8232 . 2 inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, 𝑆)
63, 4, 53eqtr4g 2669 1 (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ◡ccnv 5037  supcsup 8229  infcinf 8230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-in 3547  df-ss 3554  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-sup 8231  df-inf 8232 This theorem is referenced by: (None)
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