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Metamath Proof Explorer

Theorem 19.38a

Description: Under a nonfreeness hypothesis, the implication 19.38 can be strengthened to an equivalence. See also 19.38b . (Contributed by BJ, 3-Nov-2021) (Proof shortened by Wolf Lammen, 9-Jul-2022)

Ref Expression
Assertion 19.38a 
|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 19.38 
 |-  ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
2 id 
 |-  ( F/ x ph -> F/ x ph )
3 2 nfrd 
 |-  ( F/ x ph -> ( E. x ph -> A. x ph ) )
4 alim 
 |-  ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
5 3 4 syl9 
 |-  ( F/ x ph -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
6 1 5 impbid2 
 |-  ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )