Theorem infeq1i 9470

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 9468	. 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 1542  infcinf 9433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3955  df-ss 3965  df-uni 4909  df-sup 9434  df-inf 9435

This theorem is referenced by:  infsn  9497  nninf  12910  nn0inf  12911  lcmcom  16527  lcmass  16548  lcmf0  16568  imasdsval2  17459  imasdsf1olem  23871  ftalem6  26572  aks4d1  40943  sticksstones2  40952  supminfxr2  44166  limsup0  44397  limsupvaluz  44411  limsupmnflem  44423  limsupvaluz2  44441  limsup10ex  44476  cnrefiisp  44533  ioodvbdlimc1lem2  44635  ioodvbdlimc2lem  44637  elaa2  44937  etransc  44986  ioorrnopn  45008  ovnval2  45248  ovolval3  45350  vonioolem2  45384