Theorem nfimt 1821

Description: Closed form of nfim 1825 and curried (exported) form of nfimt2 1822. (Contributed by BJ, 20-Oct-2021.)

Assertion

Ref	Expression
nfimt	|- ( F/ x ph -> ( F/ x ps -> F/ x ( ph -> ps ) ) )

Proof of Theorem nfimt

Step	Hyp	Ref	Expression
1		19.35 1805	. . 3 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
2		df-nf 1710	. . . . . . 7 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
3	2	biimpi 206	. . . . . 6 |- ( F/ x ph -> ( E. x ph -> A. x ph ) )
4	3	imim1d 82	. . . . 5 |- ( F/ x ph -> ( ( A. x ph -> E. x ps ) -> ( E. x ph -> E. x ps ) ) )
5		df-nf 1710	. . . . . . 7 |- ( F/ x ps <-> ( E. x ps -> A. x ps ) )
6	5	biimpi 206	. . . . . 6 |- ( F/ x ps -> ( E. x ps -> A. x ps ) )
7	6	imim2d 57	. . . . 5 |- ( F/ x ps -> ( ( E. x ph -> E. x ps ) -> ( E. x ph -> A. x ps ) ) )
8	4, 7	syl9 77	. . . 4 |- ( F/ x ph -> ( F/ x ps -> ( ( A. x ph -> E. x ps ) -> ( E. x ph -> A. x ps ) ) ) )
9		19.38 1766	. . . 4 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
10	8, 9	syl8 76	. . 3 |- ( F/ x ph -> ( F/ x ps -> ( ( A. x ph -> E. x ps ) -> A. x ( ph -> ps ) ) ) )
11	1, 10	syl7bi 245	. 2 |- ( F/ x ph -> ( F/ x ps -> ( E. x ( ph -> ps ) -> A. x ( ph -> ps ) ) ) )
12		df-nf 1710	. 2 |- ( F/ x ( ph -> ps ) <-> ( E. x ( ph -> ps ) -> A. x ( ph -> ps ) ) )
13	11, 12	syl6ibr 242	1 |- ( F/ x ph -> ( F/ x ps -> F/ x ( ph -> ps ) ) )

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

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

This theorem is referenced by:  nfimt2  1822  bj-dvelimdv1  32835