Theorem infeq123d 7179

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

Hypotheses

Ref	Expression
infeq123d.a	|- ( ph -> A = D )
infeq123d.b	|- ( ph -> B = E )
infeq123d.c	|- ( ph -> C = F )

Assertion

Ref	Expression
infeq123d	|- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Proof of Theorem infeq123d

Step	Hyp	Ref	Expression
1		infeq123d.a	. . 3 |- ( ph -> A = D )
2		infeq123d.b	. . 3 |- ( ph -> B = E )
3		infeq123d.c	. . . 4 |- ( ph -> C = F )
4	3	cnveqd 4897	. . 3 |- ( ph -> `' C = `' F )
5	1, 2, 4	supeq123d 7154	. 2 |- ( ph -> sup ( A , B , `' C ) = sup ( D , E , `' F ) )
6		df-inf 7148	. 2 |- inf ( A , B , C ) = sup ( A , B , `' C )
7		df-inf 7148	. 2 |- inf ( D , E , F ) = sup ( D , E , `' F )
8	5, 6, 7	3eqtr4g 2287	1 |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Syntax hints:   -> wi 4   = wceq 1395   `'ccnv 4717   supcsup 7145  infcinf 7146

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-in 3203  df-ss 3210  df-uni 3888  df-br 4083  df-opab 4145  df-cnv 4726  df-sup 7147  df-inf 7148

This theorem is referenced by: (None)