Theorem infeq1 6976

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

Assertion

Ref	Expression
infeq1	|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Proof of Theorem infeq1

Step	Hyp	Ref	Expression
1		supeq1 6951	. 2 |- ( B = C -> sup ( B , A , `' R ) = sup ( C , A , `' R ) )
2		df-inf 6950	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
3		df-inf 6950	. 2 |- inf ( C , A , R ) = sup ( C , A , `' R )
4	1, 2, 3	3eqtr4g 2224	1 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1343   `'ccnv 4603   supcsup 6947  infcinf 6948

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-uni 3790  df-sup 6949  df-inf 6950

This theorem is referenced by:  infeq1d  6977  infeq1i  6978