Metamath Proof Explorer

Theorem 19.43

Description: Theorem 19.43 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 27-Jun-2014)

Ref Expression
Assertion 19.43 
|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )

Proof

Step Hyp Ref Expression
1 df-or 
 |-  ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
2 1 exbii 
 |-  ( E. x ( ph \/ ps ) <-> E. x ( -. ph -> ps ) )
3 19.35 
 |-  ( E. x ( -. ph -> ps ) <-> ( A. x -. ph -> E. x ps ) )
4 alnex 
 |-  ( A. x -. ph <-> -. E. x ph )
5 4 imbi1i 
 |-  ( ( A. x -. ph -> E. x ps ) <-> ( -. E. x ph -> E. x ps ) )
6 2 3 5 3bitri 
 |-  ( E. x ( ph \/ ps ) <-> ( -. E. x ph -> E. x ps ) )
7 df-or 
 |-  ( ( E. x ph \/ E. x ps ) <-> ( -. E. x ph -> E. x ps ) )
8 6 7 bitr4i 
 |-  ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )