Theorem nfbidf 1533

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem nfbidf

Step	Hyp	Ref	Expression
1		nfbidf.1	. . . 4 |- F/ x ph
2	1	nfri 1513	. . 3 |- ( ph -> A. x ph )
3		nfbidf.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
4	2, 3	albidh 1474	. . . 4 |- ( ph -> ( A. x ps <-> A. x ch ) )
5	3, 4	imbi12d 233	. . 3 |- ( ph -> ( ( ps -> A. x ps ) <-> ( ch -> A. x ch ) ) )
6	2, 5	albidh 1474	. 2 |- ( ph -> ( A. x ( ps -> A. x ps ) <-> A. x ( ch -> A. x ch ) ) )
7		df-nf 1455	. 2 |- ( F/ x ps <-> A. x ( ps -> A. x ps ) )
8		df-nf 1455	. 2 |- ( F/ x ch <-> A. x ( ch -> A. x ch ) )
9	6, 7, 8	3bitr4g 222	1 |- ( ph -> ( F/ x ps <-> F/ x ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1347   F/wnf 1454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-gen 1443  ax-4 1504

This theorem depends on definitions:  df-bi 116  df-nf 1455

This theorem is referenced by:  dvelimdf  2010  nfcjust  2301  nfceqdf  2312  nfabdw  2332