Theorem infeq1 7178

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

Assertion

Ref	Expression
infeq1	|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Proof of Theorem infeq1

Step	Hyp	Ref	Expression
1		supeq1 7153	. 2 |- ( B = C -> sup ( B , A , `' R ) = sup ( C , A , `' R ) )
2		df-inf 7152	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
3		df-inf 7152	. 2 |- inf ( C , A , R ) = sup ( C , A , `' R )
4	1, 2, 3	3eqtr4g 2287	1 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1395   `'ccnv 4718   supcsup 7149  infcinf 7150

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-uni 3889  df-sup 7151  df-inf 7152

This theorem is referenced by:  infeq1d  7179  infeq1i  7180