Theorem nfinf 7321

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 7289	. 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 4939	. . 3 |- F/_ x `' R
6	2, 3, 5	nfsup 7296	. 2 |- F/_ x sup ( A , B , `' R )
7	1, 6	nfcxfr 2383	1 |- F/_ xinf ( A , B , R )

Syntax hints:   F/_wnfc 2373   `'ccnv 4753   supcsup 7286  infcinf 7287

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-cnv 4762  df-sup 7288  df-inf 7289

This theorem is referenced by: (None)