Theorem eupickbi 2057

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 2055	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
2	1	ex 114	. 2 |- ( E! x ph -> ( E. x ( ph /\ ps ) -> A. x ( ph -> ps ) ) )
3		hba1 1503	. . . . 5 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
4		ancl 314	. . . . . . 7 |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) )
5		simpl 108	. . . . . . 7 |- ( ( ph /\ ps ) -> ph )
6	4, 5	impbid1 141	. . . . . 6 |- ( ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
7	6	sps 1500	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
8	3, 7	eubidh 1981	. . . 4 |- ( A. x ( ph -> ps ) -> ( E! x ph <-> E! x ( ph /\ ps ) ) )
9		euex 2005	. . . 4 |- ( E! x ( ph /\ ps ) -> E. x ( ph /\ ps ) )
10	8, 9	syl6bi 162	. . 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 128	1 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1312   E.wex 1451   E!weu 1975

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979

This theorem is referenced by: (None)