Theorem nfcnv 4806

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 4634	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2319	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2319	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4049	. . 3 |- F/ x z A y
6	5	nfopab 4071	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2316	1 |- F/_ x `' A

Syntax hints:   F/_wnfc 2306   class class class wbr 4003   {copab 4063   `'ccnv 4625

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634

This theorem is referenced by:  nfrn  4872  nffun  5239  nff1  5419  nfinf  7015