Theorem eupickbi 2169

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 2167	. . 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 1593	. . . . 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 1590	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ph <-> ( ph /\ ps ) ) )
8	3, 7	eubidh 2092	. . . 4 |- ( A. x ( ph -> ps ) -> ( E! x ph <-> E! x ( ph /\ ps ) ) )
9		euex 2116	. . . 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 1400   E.wex 1545   E!weu 2086

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)