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Metamath Proof Explorer

Theorem bj-nfimt

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

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

Proof

Step Hyp Ref Expression
1 19.35 
 |-  ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
2 id 
 |-  ( F/ x ph -> F/ x ph )
3 2 nfrd 
 |-  ( F/ x ph -> ( E. x ph -> A. x ph ) )
4 3 imim1d 
 |-  ( F/ x ph -> ( ( A. x ph -> E. x ps ) -> ( E. x ph -> E. x ps ) ) )
5 1 4 biimtrid 
 |-  ( F/ x ph -> ( E. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) ) )
6 id 
 |-  ( F/ x ps -> F/ x ps )
7 6 nfrd 
 |-  ( F/ x ps -> ( E. x ps -> A. x ps ) )
8 7 imim2d 
 |-  ( F/ x ps -> ( ( E. x ph -> E. x ps ) -> ( E. x ph -> A. x ps ) ) )
9 19.38 
 |-  ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
10 8 9 syl6 
 |-  ( F/ x ps -> ( ( E. x ph -> E. x ps ) -> A. x ( ph -> ps ) ) )
11 5 10 syl9 
 |-  ( F/ x ph -> ( F/ x ps -> ( E. x ( ph -> ps ) -> A. x ( ph -> ps ) ) ) )
12 df-nf 
 |-  ( F/ x ( ph -> ps ) <-> ( E. x ( ph -> ps ) -> A. x ( ph -> ps ) ) )
13 11 12 imbitrrdi 
 |-  ( F/ x ph -> ( F/ x ps -> F/ x ( ph -> ps ) ) )