Theorem infeq2 8385

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

Assertion

Ref	Expression
infeq2	|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Proof of Theorem infeq2

Step	Hyp	Ref	Expression
1		supeq2 8354	. 2 |- ( B = C -> sup ( A , B , `' R ) = sup ( A , C , `' R ) )
2		df-inf 8349	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
3		df-inf 8349	. 2 |- inf ( A , C , R ) = sup ( A , C , `' R )
4	1, 2, 3	3eqtr4g 2681	1 |- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )

Syntax hints:   -> wi 4   = wceq 1483   `'ccnv 5113   supcsup 8346  infcinf 8347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-uni 4437  df-sup 8348  df-inf 8349

This theorem is referenced by: (None)