Theorem eupickbi 16400

Description: Theorem *14.26 in [WhiteheadRussell] p. 192.

Assertion

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

Proof of Theorem eupickbi

Step	Hyp	Ref	Expression
1		eupicka 1835	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> A.x(ph -> ps))
2	1	ex 402	. 2 |- (E!xph -> (E.x(ph /\ ps) -> A.x(ph -> ps)))
3		hba1 1350	. . . . . 6 |- (A.x(ph -> ps) -> A.xA.x(ph -> ps))
4		ancl 318	. . . . . . . 8 |- ((ph -> ps) -> (ph -> (ph /\ ps)))
5		simpl 346	. . . . . . . 8 |- ((ph /\ ps) -> ph)
6	4, 5	impbid1 575	. . . . . . 7 |- ((ph -> ps) -> (ph <-> (ph /\ ps)))
7	6	a4s 1330	. . . . . 6 |- (A.x(ph -> ps) -> (ph <-> (ph /\ ps)))
8	3, 7	eubid 1778	. . . . 5 |- (A.x(ph -> ps) -> (E!xph <-> E!x(ph /\ ps)))
9	8	biimpd 170	. . . 4 |- (A.x(ph -> ps) -> (E!xph -> E!x(ph /\ ps)))
10		euex 1788	. . . 4 |- (E!x(ph /\ ps) -> E.x(ph /\ ps))
11	9, 10	syl6 25	. . 3 |- (A.x(ph -> ps) -> (E!xph -> E.x(ph /\ ps)))
12	11	com12 14	. 2 |- (E!xph -> (A.x(ph -> ps) -> E.x(ph /\ ps)))
13	2, 12	impbid 574	1 |- (E!xph -> (E.x(ph /\ ps) <-> A.x(ph -> ps)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296  E.wex 1326  E!weu 1771

This theorem is referenced by:  sbaniota 16401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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