Theorem infeq1i 8425

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 8423	. 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 1523  infcinf 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-uni 4469  df-sup 8389  df-inf 8390

This theorem is referenced by:  infsn  8451  nninf  11807  nn0inf  11808  lcmcom  15353  lcmass  15374  lcmf0  15394  imasdsval2  16223  imasdsf1olem  22225  ftalem6  24849  supminfxr2  40012  limsup0  40244  limsupvaluz  40258  limsupmnflem  40270  limsupvaluz2  40288  limsup10ex  40323  cnrefiisp  40374  ioodvbdlimc1lem2  40465  ioodvbdlimc2lem  40467  elaa2  40769  etransc  40818  ioorrnopn  40843  ovnval2  41080  ovolval3  41182  vonioolem2  41216