Theorem nfinf 7357

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 7325	. 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 4959	. . 3 |- F/_ x `' R
6	2, 3, 5	nfsup 7332	. 2 |- F/_ x sup ( A , B , `' R )
7	1, 6	nfcxfr 2389	1 |- F/_ xinf ( A , B , R )

This proof depends on syntax axioms:   F/_wnfc 2379   `'ccnv 4773   supcsup 7322  infcinf 7323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-cnv 4782  df-sup 7324  df-inf 7325

This theorem is used by: (None)