Theorem nfnfc 2585

Description: Hypothesis builder for F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfnfc.1	|- F/_ x A

Assertion

Ref	Expression
nfnfc	|- F/ x F/_ y A

Proof of Theorem nfnfc

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2568	. 2 |- ( F/_ y A <-> A. z F/ y z e. A )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2573	. . . 4 |- F/ x z e. A
4	3	nfnf 1875	. . 3 |- F/ x F/ y z e. A
5	4	nfal 1873	. 2 |- F/ x A. z F/ y z e. A
6	1, 5	nfxfr 1615	1 |- F/ x F/_ y A

Syntax hints:   A.wal 1367   F/wnf 1589   e. wcel 1756   F/_wnfc 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-cleq 2436  df-clel 2439  df-nfc 2568

This theorem is referenced by: (None)