Metamath Proof Explorer

Theorem eupickbi

Description: Theorem *14.26 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 27-Dec-2018)

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

Proof

Step Hyp Ref Expression
1 eupicka 
 |-  ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
2 1 ex 
 |-  ( E! x ph -> ( E. x ( ph /\ ps ) -> A. x ( ph -> ps ) ) )
3 euex 
 |-  ( E! x ph -> E. x ph )
4 exintr 
 |-  ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
5 3 4 syl5com 
 |-  ( E! x ph -> ( A. x ( ph -> ps ) -> E. x ( ph /\ ps ) ) )
6 2 5 impbid 
 |-  ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )