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Theorem infeq1d 8540
Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypothesis
Ref Expression
infeq1d.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
infeq1d (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Proof of Theorem infeq1d
StepHypRef Expression
1 infeq1d.1 . 2 (𝜑𝐵 = 𝐶)
2 infeq1 8539 . 2 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
31, 2syl 17 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  infcinf 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-uni 4581  df-sup 8505  df-inf 8506
This theorem is referenced by:  limsupval  14396  lcmval  15499  lcmass  15521  lcmfval  15528  lcmf0val  15529  lcmfpr  15534  odzval  15690  ramval  15906  imasval  16365  imasdsval  16369  gexval  18185  nmofval  22711  nmoval  22712  metdsval  22843  lebnumlem1  22953  lebnumlem3  22955  ovolval  23434  ovolshft  23471  ioorf  23533  mbflimsup  23624  ig1pval  24123  elqaalem1  24265  elqaalem2  24266  elqaalem3  24267  elqaa  24268  omsval  30656  omsfval  30657  ballotlemi  30863  pellfundval  37938  dgraaval  38208  supminfrnmpt  40162  infxrpnf  40164  infxrpnf2  40183  supminfxr  40184  supminfxr2  40189  supminfxrrnmpt  40191  limsupval3  40419  limsupresre  40423  limsupresico  40427  limsuppnfdlem  40428  limsupvaluz  40435  limsupvaluzmpt  40444  liminfval  40486  liminfgval  40489  liminfval5  40492  limsupresxr  40493  liminfresxr  40494  liminfval2  40495  liminfresico  40498  liminf10ex  40501  liminfvalxr  40510  fourierdlem31  40850  ovnval  41253  ovnval2  41257  ovnval2b  41264  ovolval2  41356  ovnovollem3  41370  smfinf  41522  smfinfmpt  41523  prmdvdsfmtnof1  42001
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