Theorem infeq1i 9440

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 9438	. 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 1570  infcinf 9402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-ss 3923  df-uni 4874  df-sup 9403  df-inf 9404

This theorem is referenced by:  infsn  9468  nninf  12954  nn0inf  12955  lcmcom  16652  lcmass  16673  lcmf0  16693  imasdsval2  17571  imasdsf1olem  24511  ftalem6  27223  aks4d1  42837  sticksstones2  42895  supminfxr2  46166  limsup0  46391  limsupvaluz  46405  limsupmnflem  46417  limsupvaluz2  46435  limsup10ex  46470  cnrefiisp  46527  ioodvbdlimc1lem2  46629  ioodvbdlimc2lem  46631  elaa2  46931  etransc  46980  ioorrnopn  47002  ovnval2  47242  ovolval3  47344  vonioolem2  47378