Theorem nfcnv 4915

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 4739	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2375	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2375	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4140	. . 3 |- F/ x z A y
6	5	nfopab 4162	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2372	1 |- F/_ x `' A

Syntax hints:   F/_wnfc 2362   class class class wbr 4093   {copab 4154   `'ccnv 4730

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 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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739

This theorem is referenced by:  nfrn  4983  nffun  5356  nff1  5549  nfinf  7276