Theorem eupickbi 1982

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 1980	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
2	1	ex 108	. 2 |- ( E! x ph -> ( E. x ( ph /\ ps ) -> A. x ( ph -> ps ) ) )
3		hba1 1433	. . . . 5 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
4		ancl 301	. . . . . . 7 |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) )
5		simpl 102	. . . . . . 7 |- ( ( ph /\ ps ) -> ph )
6	4, 5	impbid1 130	. . . . . 6 |- ( ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
7	6	sps 1430	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
8	3, 7	eubidh 1906	. . . 4 |- ( A. x ( ph -> ps ) -> ( E! x ph <-> E! x ( ph /\ ps ) ) )
9		euex 1930	. . . 4 |- ( E! x ( ph /\ ps ) -> E. x ( ph /\ ps ) )
10	8, 9	syl6bi 152	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ph -> E. x ( ph /\ ps ) ) )
11	10	com12 27	. 2 |- ( E! x ph -> ( A. x ( ph -> ps ) -> E. x ( ph /\ ps ) ) )
12	2, 11	impbid 120	1 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by: (None)