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Metamath Proof Explorer

Theorem bj-19.21t

Description: Statement 19.21t proved from modalK (obsoleting 19.21v ). (Contributed by BJ, 2-Dec-2023)

Ref Expression
Assertion bj-19.21t 
|- ( F// x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Proof

Step Hyp Ref Expression
1 bj-nnf-alrim 
 |-  ( F// x ph -> ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) )
2 bj-nnfe 
 |-  ( F// x ph -> ( E. x ph -> ph ) )
3 2 imim1d 
 |-  ( F// x ph -> ( ( ph -> A. x ps ) -> ( E. x ph -> A. x ps ) ) )
4 19.38 
 |-  ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
5 3 4 syl6 
 |-  ( F// x ph -> ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) ) )
6 1 5 impbid 
 |-  ( F// x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )