Theorem nfcnv 4959

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 4782	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2392	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2392	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4177	. . 3 |- F/ x z A y
6	5	nfopab 4199	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2389	1 |- F/_ x `' A

This proof depends on syntax axioms:   F/_wnfc 2379   class class class wbr 4130   {copab 4191   `'ccnv 4773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782

This theorem is used by:  nfrn  5027  nffun  5400  nff1  5596  nfinf  7357