Theorem nfinf 6994

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 6962	. 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 4790	. . 3 |- F/_ x `' R
6	2, 3, 5	nfsup 6969	. 2 |- F/_ x sup ( A , B , `' R )
7	1, 6	nfcxfr 2309	1 |- F/_ xinf ( A , B , R )

Syntax hints:   F/_wnfc 2299   `'ccnv 4610   supcsup 6959  infcinf 6960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-cnv 4619  df-sup 6961  df-inf 6962

This theorem is referenced by: (None)