Theorem nfnfc 2573

Description: Hypothesis builder for F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2026. (Revised by Wolf Lammen, 10-Dec-2019.)

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 2552	. 2 |- ( F/_ y A <-> A. z F/ y z e. A )
2		nfnfc.1	. . . . 5 |- F/_ x A
3		nfcr 2555	. . . . 5 |- ( F/_ x A -> F/ x z e. A )
4	2, 3	ax-mp 5	. . . 4 |- F/ x z e. A
5	4	nfnf 1977	. . 3 |- F/ x F/ y z e. A
6	5	nfal 1975	. 2 |- F/ x A. z F/ y z e. A
7	1, 6	nfxfr 1666	1 |- F/ x F/_ y A

Syntax hints:   A.wal 1403   F/wnf 1637   e. wcel 1842   F/_wnfc 2550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878

This theorem depends on definitions:  df-bi 185  df-ex 1634  df-nf 1638  df-nfc 2552

This theorem is referenced by: (None)