Theorem eupickbi 2160

Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Proof of Theorem eupickbi

Step	Hyp	Ref	Expression
1		eupicka 2158	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
2	1	ex 115	. 2 |- ( E! x ph -> ( E. x ( ph /\ ps ) -> A. x ( ph -> ps ) ) )
3		hba1 1586	. . . . 5 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
4		ancl 318	. . . . . . 7 |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) )
5		simpl 109	. . . . . . 7 |- ( ( ph /\ ps ) -> ph )
6	4, 5	impbid1 142	. . . . . 6 |- ( ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
7	6	sps 1583	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
8	3, 7	eubidh 2083	. . . 4 |- ( A. x ( ph -> ps ) -> ( E! x ph <-> E! x ( ph /\ ps ) ) )
9		euex 2107	. . . 4 |- ( E! x ( ph /\ ps ) -> E. x ( ph /\ ps ) )
10	8, 9	biimtrdi 163	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ph -> E. x ( ph /\ ps ) ) )
11	10	com12 30	. 2 |- ( E! x ph -> ( A. x ( ph -> ps ) -> E. x ( ph /\ ps ) ) )
12	2, 11	impbid 129	1 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   E.wex 1538   E!weu 2077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by: (None)