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Metamath Proof Explorer

Theorem 19.35

Description: Theorem 19.35 of Margaris p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 27-Jun-2014)

Ref Expression
Assertion 19.35 
|- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )

Proof

Step Hyp Ref Expression
1 pm2.27 
 |-  ( ph -> ( ( ph -> ps ) -> ps ) )
2 1 aleximi 
 |-  ( A. x ph -> ( E. x ( ph -> ps ) -> E. x ps ) )
3 2 com12 
 |-  ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
4 exnal 
 |-  ( E. x -. ph <-> -. A. x ph )
5 pm2.21 
 |-  ( -. ph -> ( ph -> ps ) )
6 5 eximi 
 |-  ( E. x -. ph -> E. x ( ph -> ps ) )
7 4 6 sylbir 
 |-  ( -. A. x ph -> E. x ( ph -> ps ) )
8 exa1 
 |-  ( E. x ps -> E. x ( ph -> ps ) )
9 7 8 ja 
 |-  ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) )
10 3 9 impbii 
 |-  ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )