Metamath Proof Explorer

Theorem 19.30

Description: Theorem 19.30 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Assertion 19.30 
|- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )

Proof

Step Hyp Ref Expression
1 exnal 
 |-  ( E. x -. ph <-> -. A. x ph )
2 pm2.53 
 |-  ( ( ph \/ ps ) -> ( -. ph -> ps ) )
3 2 aleximi 
 |-  ( A. x ( ph \/ ps ) -> ( E. x -. ph -> E. x ps ) )
4 1 3 syl5bir 
 |-  ( A. x ( ph \/ ps ) -> ( -. A. x ph -> E. x ps ) )
5 4 orrd 
 |-  ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )