Theorem nfcnv 4790

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 4619	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2312	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2312	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4035	. . 3 |- F/ x z A y
6	5	nfopab 4057	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2309	1 |- F/_ x `' A

Syntax hints:   F/_wnfc 2299   class class class wbr 3989   {copab 4049   `'ccnv 4610

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-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-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by:  nfrn  4856  nffun  5221  nff1  5401  nfinf  6994