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Theorem infeq3 6708
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 4610 . . 3 (𝑅 = 𝑆𝑅 = 𝑆)
2 supeq3 6683 . . 3 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
31, 2syl 14 . 2 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
4 df-inf 6678 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
5 df-inf 6678 . 2 inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, 𝑆)
63, 4, 53eqtr4g 2145 1 (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  ccnv 4437  supcsup 6675  infcinf 6676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-in 3005  df-ss 3012  df-uni 3654  df-br 3846  df-opab 3900  df-cnv 4446  df-sup 6677  df-inf 6678
This theorem is referenced by: (None)
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