Theorem infeq1i 7072

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 7070	. 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 1364  infcinf 7042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-uni 3836  df-sup 7043  df-inf 7044

This theorem is referenced by:  mincom  11372  xrbdtri  11419  nninfctlemfo  12177  lcmcom  12202  lcmass  12223