Theorem infeq2 7073

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 7048	. 2 |- ( B = C -> sup ( A , B , `' R ) = sup ( A , C , `' R ) )
2		df-inf 7044	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
3		df-inf 7044	. 2 |- inf ( A , C , R ) = sup ( A , C , `' R )
4	1, 2, 3	3eqtr4g 2251	1 |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Syntax hints:   -> wi 4   = wceq 1364   `'ccnv 4658   supcsup 7041  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: (None)