Theorem nfinf 7119

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 7087	. 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 4857	. . 3 |- F/_ x `' R
6	2, 3, 5	nfsup 7094	. 2 |- F/_ x sup ( A , B , `' R )
7	1, 6	nfcxfr 2345	1 |- F/_ xinf ( A , B , R )

Syntax hints:   F/_wnfc 2335   `'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-3an 983  df-tru 1376  df-fal 1379  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-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-cnv 4683  df-sup 7086  df-inf 7087

This theorem is referenced by: (None)