Theorem infeq3 7143

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 4870	. . 3 |- ( R = S -> `' R = `' S )
2		supeq3 7118	. . 3 |- ( `' R = `' S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
3	1, 2	syl 14	. 2 |- ( R = S -> sup ( A , B , `' R ) = sup ( A , B , `' S ) )
4		df-inf 7113	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
5		df-inf 7113	. 2 |- inf ( A , B , S ) = sup ( A , B , `' S )
6	3, 4, 5	3eqtr4g 2265	1 |- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4692   supcsup 7110  infcinf 7111

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-in 3180  df-ss 3187  df-uni 3865  df-br 4060  df-opab 4122  df-cnv 4701  df-sup 7112  df-inf 7113

This theorem is referenced by: (None)