Theorem infeq2 7218

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 7193	. 2 |- ( B = C -> sup ( A , B , `' R ) = sup ( A , C , `' R ) )
2		df-inf 7189	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
3		df-inf 7189	. 2 |- inf ( A , C , R ) = sup ( A , C , `' R )
4	1, 2, 3	3eqtr4g 2288	1 |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Syntax hints:   -> wi 4   = wceq 1397   `'ccnv 4726   supcsup 7186  infcinf 7187

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-uni 3895  df-sup 7188  df-inf 7189

This theorem is referenced by: (None)