Theorem nfcnv 5301

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 5122	. 2 |- `' A = { <. y , z >. | z A y }
2		nfcv 2764	. . . 4 |- F/_ x z
3		nfcnv.1	. . . 4 |- F/_ x A
4		nfcv 2764	. . . 4 |- F/_ x y
5	2, 3, 4	nfbr 4699	. . 3 |- F/ x z A y
6	5	nfopab 4718	. 2 |- F/_ x { <. y , z >. | z A y }
7	1, 6	nfcxfr 2762	1 |- F/_ x `' A

Syntax hints:   F/_wnfc 2751   class class class wbr 4653   {copab 4712   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122

This theorem is referenced by:  nfrn  5368  nfpred  5685  nffun  5911  nff1  6099  nfsup  8357  nfinf  8388  gsumcom2  18374  ptbasfi  21384  mbfposr  23419  itg1climres  23481  funcnvmptOLD  29467  funcnvmpt  29468  nfwsuc  31764  aomclem8  37631  rfcnpre1  39178  rfcnpre2  39190  smfpimcc  41014