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Metamath Proof Explorer

Theorem 19.26

Description: Theorem 19.26 of Margaris p. 90. Also Theorem *10.22 of WhiteheadRussell p. 147. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 4-Jul-2014)

Ref Expression
Assertion 19.26 
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( ph /\ ps ) -> ph )
2 1 alimi 
 |-  ( A. x ( ph /\ ps ) -> A. x ph )
3 simpr 
 |-  ( ( ph /\ ps ) -> ps )
4 3 alimi 
 |-  ( A. x ( ph /\ ps ) -> A. x ps )
5 2 4 jca 
 |-  ( A. x ( ph /\ ps ) -> ( A. x ph /\ A. x ps ) )
6 id 
 |-  ( ( ph /\ ps ) -> ( ph /\ ps ) )
7 6 alanimi 
 |-  ( ( A. x ph /\ A. x ps ) -> A. x ( ph /\ ps ) )
8 5 7 impbii 
 |-  ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )