Theorem infeq1d 8943

Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)

Hypothesis

Ref	Expression
infeq1d.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
infeq1d	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Proof of Theorem infeq1d

Step	Hyp	Ref	Expression
1		infeq1d.1	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
2		infeq1 8942	. 2 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
3	1, 2	syl 17	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Syntax hints:   → wi 4   = wceq 1537  infcinf 8907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-in 3945  df-ss 3954  df-uni 4841  df-sup 8908  df-inf 8909

This theorem is referenced by:  limsupval  14833  lcmval  15938  lcmass  15960  lcmfval  15967  lcmf0val  15968  lcmfpr  15973  odzval  16130  ramval  16346  imasval  16786  imasdsval  16790  gexval  18705  nmofval  23325  nmoval  23326  metdsval  23457  lebnumlem1  23567  lebnumlem3  23569  ovolval  24076  ovolshft  24114  ioorf  24176  mbflimsup  24269  ig1pval  24768  elqaalem1  24910  elqaalem2  24911  elqaalem3  24912  elqaa  24913  omsval  31553  omsfval  31554  ballotlemi  31760  pellfundval  39484  dgraaval  39751  supminfrnmpt  41726  infxrpnf  41728  infxrpnf2  41746  supminfxr  41747  supminfxr2  41752  supminfxrrnmpt  41754  limsupval3  41980  limsupresre  41984  limsupresico  41988  limsuppnfdlem  41989  limsupvaluz  41996  limsupvaluzmpt  42005  liminfval  42047  liminfgval  42050  liminfval5  42053  limsupresxr  42054  liminfresxr  42055  liminfval2  42056  liminfresico  42059  liminf10ex  42062  liminfvalxr  42071  fourierdlem31  42430  ovnval  42830  ovnval2  42834  ovnval2b  42841  ovolval2  42933  ovnovollem3  42947  smfinf  43099  smfinfmpt  43100  prmdvdsfmtnof1  43756