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)

This proof depends on syntax axioms:   = wceq 1570  infcinf 9463

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-uni 4871  df-sup 9464  df-inf 9465

This theorem is used by:  infsn  9529  nninf  13086  nn0inf  13087  lcmcom  16800  lcmass  16821  lcmf0  16841  imasdsval2  17720  imasdsf1olem  24725  ftalem6  27438  aks4d1  43342  sticksstones2  43400  supminfxr2  46694  limsup0  46919  limsupvaluz  46933  limsupmnflem  46945  limsupvaluz2  46963  limsup10ex  46998  cnrefiisp  47055  ioodvbdlimc1lem2  47157  ioodvbdlimc2lem  47159  elaa2  47459  etransc  47508  ioorrnopn  47530  ovnval2  47770  ovolval3  47872  vonioolem2  47906