Theorem infeq3 6902

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 4713	. . 3 |- ( R = S -> `' R = `' S )
2		supeq3 6877	. . 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 6872	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
5		df-inf 6872	. 2 |- inf ( A , B , S ) = sup ( A , B , `' S )
6	3, 4, 5	3eqtr4g 2197	1 |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

Syntax hints:   -> wi 4   = wceq 1331   `'ccnv 4538   supcsup 6869  infcinf 6870

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-in 3077  df-ss 3084  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-sup 6871  df-inf 6872

This theorem is referenced by: (None)