Theorem pm10.55 38568

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

Assertion

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

Proof of Theorem pm10.55

Step	Hyp	Ref	Expression
1		exsimpl 1795	. . 3 |- ( E. x ( ph /\ ps ) -> E. x ph )
2	1	anim1i 592	. 2 |- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) -> ( E. x ph /\ A. x ( ph -> ps ) ) )
3		exintr 1819	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
4	3	imdistanri 727	. 2 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) )
5	2, 4	impbii 199	1 |- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) <-> ( E. x ph /\ A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)