Theorem nfcnv 4875

Description: Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	|- F/_ x A

Assertion

Ref	Expression
nfcnv	|- F/_ x `' A

Proof of Theorem nfcnv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 4701	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2350	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2350	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4106	. . 3 |- F/ x z A y
6	5	nfopab 4128	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2347	1 |- F/_ x `' A

Syntax hints:   F/_wnfc 2337   class class class wbr 4059   {copab 4120   `'ccnv 4692

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701

This theorem is referenced by:  nfrn  4942  nffun  5313  nff1  5501  nfinf  7145