Theorem infeq1i 8384

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 8382	. 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 1483  infcinf 8347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-uni 4437  df-sup 8348  df-inf 8349

This theorem is referenced by:  infsn  8410  nninf  11769  nn0inf  11770  lcmcom  15306  lcmass  15327  lcmf0  15347  imasdsval2  16176  imasdsf1olem  22178  ftalem6  24804  supminfxr2  39699  limsup0  39926  limsupvaluz  39940  limsupmnflem  39952  limsupvaluz2  39970  limsup10ex  40005  cnrefiisp  40056  ioodvbdlimc1lem2  40147  ioodvbdlimc2lem  40149  elaa2  40451  etransc  40500  ioorrnopn  40525  ovnval2  40759  ovolval3  40861  vonioolem2  40895