Theorem infeq123d 7075

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 4838	. . 3 |- ( ph -> `' C = `' F )
5	1, 2, 4	supeq123d 7050	. 2 |- ( ph -> sup ( A , B , `' C ) = sup ( D , E , `' F ) )
6		df-inf 7044	. 2 |- inf ( A , B , C ) = sup ( A , B , `' C )
7		df-inf 7044	. 2 |- inf ( D , E , F ) = sup ( D , E , `' F )
8	5, 6, 7	3eqtr4g 2251	1 |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Syntax hints:   -> wi 4   = wceq 1364   `'ccnv 4658   supcsup 7041  infcinf 7042

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-in 3159  df-ss 3166  df-uni 3836  df-br 4030  df-opab 4091  df-cnv 4667  df-sup 7043  df-inf 7044

This theorem is referenced by: (None)