Theorem infeq1 6850

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 6825	. 2 |- ( B = C -> sup ( B , A , `' R ) = sup ( C , A , `' R ) )
2		df-inf 6824	. 2 |- inf ( B , A , R ) = sup ( B , A , `' R )
3		df-inf 6824	. 2 |- inf ( C , A , R ) = sup ( C , A , `' R )
4	1, 2, 3	3eqtr4g 2172	1 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1314   `'ccnv 4498   supcsup 6821  infcinf 6822

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-uni 3703  df-sup 6823  df-inf 6824

This theorem is referenced by:  infeq1d  6851  infeq1i  6852