Theorem eupickbi 2270

Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Proof of Theorem eupickbi

Step	Hyp	Ref	Expression
1		eupicka 2268	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> A.x(ph -> ps))
2	1	ex 423	. 2 |- (E!xph -> (E.x(ph /\ ps) -> A.x(ph -> ps)))
3		nfa1 1788	. . . . 5 |- F/xA.x(ph -> ps)
4		ancl 529	. . . . . . 7 |- ((ph -> ps) -> (ph -> (ph /\ ps)))
5		simpl 443	. . . . . . 7 |- ((ph /\ ps) -> ph)
6	4, 5	impbid1 194	. . . . . 6 |- ((ph -> ps) -> (ph <-> (ph /\ ps)))
7	6	sps 1754	. . . . 5 |- (A.x(ph -> ps) -> (ph <-> (ph /\ ps)))
8	3, 7	eubid 2211	. . . 4 |- (A.x(ph -> ps) -> (E!xph <-> E!x(ph /\ ps)))
9		euex 2227	. . . 4 |- (E!x(ph /\ ps) -> E.x(ph /\ ps))
10	8, 9	syl6bi 219	. . 3 |- (A.x(ph -> ps) -> (E!xph -> E.x(ph /\ ps)))
11	10	com12 27	. 2 |- (E!xph -> (A.x(ph -> ps) -> E.x(ph /\ ps)))
12	2, 11	impbid 183	1 |- (E!xph -> (E.x(ph /\ ps) <-> A.x(ph -> ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)