Theorem infeq1i 9501

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

Hypothesis

Ref	Expression
infeq1i.1	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
infeq1i	⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

Proof of Theorem infeq1i

Step	Hyp	Ref	Expression
1		infeq1i.1	. 2 ⊢ 𝐵 = 𝐶
2		infeq1 9499	. 2 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
3	1, 2	ax-mp 5	1 ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

Syntax hints:   = wceq 1534  infcinf 9464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-sup 9465  df-inf 9466

This theorem is referenced by:  infsn  9528  nninf  12943  nn0inf  12944  lcmcom  16563  lcmass  16584  lcmf0  16604  imasdsval2  17497  imasdsf1olem  24278  ftalem6  27009  aks4d1  41560  sticksstones2  41619  supminfxr2  44851  limsup0  45082  limsupvaluz  45096  limsupmnflem  45108  limsupvaluz2  45126  limsup10ex  45161  cnrefiisp  45218  ioodvbdlimc1lem2  45320  ioodvbdlimc2lem  45322  elaa2  45622  etransc  45671  ioorrnopn  45693  ovnval2  45933  ovolval3  46035  vonioolem2  46069