Theorem eupickbi 2023

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 2021	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
2	1	ex 113	. 2 |- ( E! x ph -> ( E. x ( ph /\ ps ) -> A. x ( ph -> ps ) ) )
3		hba1 1473	. . . . 5 |- ( A. x ( ph -> ps ) -> A. x A. x ( ph -> ps ) )
4		ancl 311	. . . . . . 7 |- ( ( ph -> ps ) -> ( ph -> ( ph /\ ps ) ) )
5		simpl 107	. . . . . . 7 |- ( ( ph /\ ps ) -> ph )
6	4, 5	impbid1 140	. . . . . 6 |- ( ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
7	6	sps 1470	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
8	3, 7	eubidh 1947	. . . 4 |- ( A. x ( ph -> ps ) -> ( E! x ph <-> E! x ( ph /\ ps ) ) )
9		euex 1971	. . . 4 |- ( E! x ( ph /\ ps ) -> E. x ( ph /\ ps ) )
10	8, 9	syl6bi 161	. . 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 127	1 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)