Theorem nfbid 1832

Description: If in a context x is not free in ps and ch, it is not free in ( ps <-> ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Hypotheses

Ref	Expression
nfbid.1	|- ( ph -> F/ x ps )
nfbid.2	|- ( ph -> F/ x ch )

Assertion

Ref	Expression
nfbid	|- ( ph -> F/ x ( ps <-> ch ) )

Proof of Theorem nfbid

Step	Hyp	Ref	Expression
1		dfbi2 660	. 2 |- ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
2		nfbid.1	. . . 4 |- ( ph -> F/ x ps )
3		nfbid.2	. . . 4 |- ( ph -> F/ x ch )
4	2, 3	nfimd 1823	. . 3 |- ( ph -> F/ x ( ps -> ch ) )
5	3, 2	nfimd 1823	. . 3 |- ( ph -> F/ x ( ch -> ps ) )
6	4, 5	nfand 1826	. 2 |- ( ph -> F/ x ( ( ps -> ch ) /\ ( ch -> ps ) ) )
7	1, 6	nfxfrd 1780	1 |- ( ph -> F/ x ( ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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

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:  nfbi  1833  nfeud2  2482  nfeqd  2772  nfiotad  5854  iota2df  5875  axextnd  9413  axrepndlem1  9414  axrepndlem2  9415  axacndlem4  9432  axacndlem5  9433  axacnd  9434  axextdist  31705  wl-eudf  33354  wl-sb8eut  33359