Theorem infeq3 7011

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 4800	. . 3 |- ( R = S -> `' R = `' S )
2		supeq3 6986	. . 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 6981	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
5		df-inf 6981	. 2 |- inf ( A , B , S ) = sup ( A , B , `' S )
6	3, 4, 5	3eqtr4g 2235	1 |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

Syntax hints:   -> wi 4   = wceq 1353   `'ccnv 4624   supcsup 6978  infcinf 6979

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-in 3135  df-ss 3142  df-uni 3810  df-br 4003  df-opab 4064  df-cnv 4633  df-sup 6980  df-inf 6981

This theorem is referenced by: (None)