Theorem infeq123d 7091

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 4843	. . 3 |- ( ph -> `' C = `' F )
5	1, 2, 4	supeq123d 7066	. 2 |- ( ph -> sup ( A , B , `' C ) = sup ( D , E , `' F ) )
6		df-inf 7060	. 2 |- inf ( A , B , C ) = sup ( A , B , `' C )
7		df-inf 7060	. 2 |- inf ( D , E , F ) = sup ( D , E , `' F )
8	5, 6, 7	3eqtr4g 2254	1 |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )

Syntax hints:   -> wi 4   = wceq 1364   `'ccnv 4663   supcsup 7057  infcinf 7058

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-in 3163  df-ss 3170  df-uni 3841  df-br 4035  df-opab 4096  df-cnv 4672  df-sup 7059  df-inf 7060

This theorem is referenced by: (None)