Theorem infeq123d 7118

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 4854	. . 3 |- ( ph -> `' C = `' F )
5	1, 2, 4	supeq123d 7093	. 2 |- ( ph -> sup ( A , B , `' C ) = sup ( D , E , `' F ) )
6		df-inf 7087	. 2 |- inf ( A , B , C ) = sup ( A , B , `' C )
7		df-inf 7087	. 2 |- inf ( D , E , F ) = sup ( D , E , `' F )
8	5, 6, 7	3eqtr4g 2263	1 |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4674   supcsup 7084  infcinf 7085

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-in 3172  df-ss 3179  df-uni 3851  df-br 4045  df-opab 4106  df-cnv 4683  df-sup 7086  df-inf 7087

This theorem is referenced by: (None)