Theorem infeq1i 9237

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 9235	. 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 1539  infcinf 9200

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-sup 9201  df-inf 9202

This theorem is referenced by:  infsn  9264  nninf  12669  nn0inf  12670  lcmcom  16298  lcmass  16319  lcmf0  16339  imasdsval2  17227  imasdsf1olem  23526  ftalem6  26227  aks4d1  40097  sticksstones2  40103  supminfxr2  43009  limsup0  43235  limsupvaluz  43249  limsupmnflem  43261  limsupvaluz2  43279  limsup10ex  43314  cnrefiisp  43371  ioodvbdlimc1lem2  43473  ioodvbdlimc2lem  43475  elaa2  43775  etransc  43824  ioorrnopn  43846  ovnval2  44083  ovolval3  44185  vonioolem2  44219