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/xps)
nfbid.2	|- (ph -> F/xch)

Assertion

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

Proof of Theorem nfbid

Step	Hyp	Ref	Expression
1		dfbi2 609	. 2 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
2		nfbid.1	. . . 4 |- (ph -> F/xps)
3		nfbid.2	. . . 4 |- (ph -> F/xch)
4	2, 3	nfimd 1808	. . 3 |- (ph -> F/x(ps -> ch))
5	3, 2	nfimd 1808	. . 3 |- (ph -> F/x(ch -> ps))
6	4, 5	nfand 1822	. 2 |- (ph -> F/x((ps -> ch) /\ (ch -> ps)))
7	1, 6	nfxfrd 1571	1 |- (ph -> F/x(ps <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfbi  1834  nfeud2  2216  nfeqd  2504  nfiotad  4343  iota2df  4366