Theorem nfnf 2158

Description: If x is not free in ph, it is not free in F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfnf.1	|- F/ x ph

Assertion

Ref	Expression
nfnf	|- F/ x F/ y ph

Proof of Theorem nfnf

Step	Hyp	Ref	Expression
1		df-nf 1710	. 2 |- ( F/ y ph <-> ( E. y ph -> A. y ph ) )
2		nfnf.1	. . . 4 |- F/ x ph
3	2	nfex 2154	. . 3 |- F/ x E. y ph
4	2	nfal 2153	. . 3 |- F/ x A. y ph
5	3, 4	nfim 1825	. 2 |- F/ x ( E. y ph -> A. y ph )
6	1, 5	nfxfr 1779	1 |- F/ x F/ y ph

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   F/wnf 1708

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-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfnfc  2774  nfnfcALT  2775  bj-nfnfc  32853  bj-nfcf  32920