Theorem infeq3 8386

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 5296	. . 3 |- ( R = S -> `' R = `' S )
2		supeq3 8355	. . 3 |- ( `' R = `' S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
3	1, 2	syl 17	. 2 |- ( R = S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
4		df-inf 8349	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
5		df-inf 8349	. 2 |- inf ( A , B , S ) = sup ( A , B , `' S )
6	3, 4, 5	3eqtr4g 2681	1 |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

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-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-sup 8348  df-inf 8349

This theorem is referenced by: (None)