Theorem nfinf 7180

Description: Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)

Hypotheses

Ref	Expression
nfinf.1	|- F/_ x A
nfinf.2	|- F/_ x B
nfinf.3	|- F/_ x R

Assertion

Ref	Expression
nfinf	|- F/_ xinf ( A , B , R )

Proof of Theorem nfinf

Step	Hyp	Ref	Expression
1		df-inf 7148	. 2 |- inf ( A , B , R ) = sup ( A , B , `' R )
2		nfinf.1	. . 3 |- F/_ x A
3		nfinf.2	. . 3 |- F/_ x B
4		nfinf.3	. . . 4 |- F/_ x R
5	4	nfcnv 4900	. . 3 |- F/_ x `' R
6	2, 3, 5	nfsup 7155	. 2 |- F/_ x sup ( A , B , `' R )
7	1, 6	nfcxfr 2369	1 |- F/_ xinf ( A , B , R )

Syntax hints:   F/_wnfc 2359   `'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-3an 1004  df-tru 1398  df-fal 1401  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-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-cnv 4726  df-sup 7147  df-inf 7148

This theorem is referenced by: (None)