Theorem infeq3 6910

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

Assertion

Ref	Expression
infeq3	|- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

Proof of Theorem infeq3

Step	Hyp	Ref	Expression
1		cnveq 4721	. . 3 |- ( R = S -> `' R = `' S )
2		supeq3 6885	. . 3 |- ( `' R = `' S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
3	1, 2	syl 14	. 2 |- ( R = S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
4		df-inf 6880	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
5		df-inf 6880	. 2 |- inf ( A , B , S ) = sup ( A , B , `' S )
6	3, 4, 5	3eqtr4g 2198	1 |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

Syntax hints:   -> wi 4   = wceq 1332   `'ccnv 4546   supcsup 6877  infcinf 6878

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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-in 3082  df-ss 3089  df-uni 3745  df-br 3938  df-opab 3998  df-cnv 4555  df-sup 6879  df-inf 6880

This theorem is referenced by: (None)