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Theorem infeq1 6706
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Proof of Theorem infeq1
StepHypRef Expression
1 supeq1 6681 . 2 (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
2 df-inf 6680 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
3 df-inf 6680 . 2 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
41, 2, 33eqtr4g 2145 1 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  ccnv 4437  supcsup 6677  infcinf 6678
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-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-uni 3654  df-sup 6679  df-inf 6680
This theorem is referenced by:  infeq1d  6707  infeq1i  6708
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