Theorem infeq123d 7346

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 4951	. . 3 |- ( ph -> `' C = `' F )
5	1, 2, 4	supeq123d 7321	. 2 |- ( ph -> sup ( A , B , `' C ) = sup ( D , E , `' F ) )
6		df-inf 7315	. 2 |- inf ( A , B , C ) = sup ( A , B , `' C )
7		df-inf 7315	. 2 |- inf ( D , E , F ) = sup ( D , E , `' F )
8	5, 6, 7	3eqtr4g 2296	1 |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Syntax hints:   -> wi 4   = wceq 1402   `'ccnv 4768   supcsup 7312  infcinf 7313

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-in 3226  df-ss 3233  df-uni 3931  df-br 4126  df-opab 4188  df-cnv 4777  df-sup 7314  df-inf 7315

This theorem is referenced by: (None)