Theorem nfbidf 2092

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

Hypotheses

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

Assertion

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

Proof of Theorem nfbidf

Step	Hyp	Ref	Expression
1		albid.1	. . . 4 |- F/ x ph
2		albid.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	1, 2	exbid 2091	. . 3 |- ( ph -> ( E. x ps <-> E. x ch ) )
4	1, 2	albid 2090	. . 3 |- ( ph -> ( A. x ps <-> A. x ch ) )
5	3, 4	imbi12d 334	. 2 |- ( ph -> ( ( E. x ps -> A. x ps ) <-> ( E. x ch -> A. x ch ) ) )
6		df-nf 1710	. 2 |- ( F/ x ps <-> ( E. x ps -> A. x ps ) )
7		df-nf 1710	. 2 |- ( F/ x ch <-> ( E. x ch -> A. x ch ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> ( F/ x ps <-> F/ x ch ) )

Syntax hints:   -> wi 4   <-> wb 196   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-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  drnf2  2330  dvelimdf  2335  nfcjust  2752  nfceqdf  2760  bj-drnf2v  32751  bj-nfcjust  32850  wl-nfimf1  33313  nfbii2  33967