Theorem pm10.55

Description: Theorem *10.55 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.55	|- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) <-> ( E. x ph /\ A. x ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		exsimpl	|- ( E. x ( ph /\ ps ) -> E. x ph )
2	1	anim1i	|- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) -> ( E. x ph /\ A. x ( ph -> ps ) ) )
3		exintr	|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
4	3	imdistanri	|- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) )
5	2 4	impbii	|- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) <-> ( E. x ph /\ A. x ( ph -> ps ) ) )