Theorem infeq1 7344

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 7319	. 2 |- ( B = C -> sup ( B , A , `' R ) = sup ( C , A , `' R ) )
2		df-inf 7318	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
3		df-inf 7318	. 2 |- inf ( C , A , R ) = sup ( C , A , `' R )
4	1, 2, 3	3eqtr4g 2296	1 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1402   `'ccnv 4771   supcsup 7315  infcinf 7316

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-uni 3934  df-sup 7317  df-inf 7318

This theorem is referenced by:  infeq1d  7345  infeq1i  7346