Theorem infeq2 6950

Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)

Assertion

Ref	Expression
infeq2	|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Proof of Theorem infeq2

Step	Hyp	Ref	Expression
1		supeq2 6925	. 2 |- ( B = C -> sup ( A , B , `' R ) = sup ( A , C , `' R ) )
2		df-inf 6921	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
3		df-inf 6921	. 2 |- inf ( A , C , R ) = sup ( A , C , `' R )
4	1, 2, 3	3eqtr4g 2215	1 |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Syntax hints:   -> wi 4   = wceq 1335   `'ccnv 4582   supcsup 6918  infcinf 6919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-uni 3773  df-sup 6920  df-inf 6921

This theorem is referenced by: (None)