Theorem infeq2 7118

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 7093	. 2 |- ( B = C -> sup ( A , B , `' R ) = sup ( A , C , `' R ) )
2		df-inf 7089	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
3		df-inf 7089	. 2 |- inf ( A , C , R ) = sup ( A , C , `' R )
4	1, 2, 3	3eqtr4g 2263	1 |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4675   supcsup 7086  infcinf 7087

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-uni 3851  df-sup 7088  df-inf 7089

This theorem is referenced by: (None)