Theorem 2eu2 2285

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu2	|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 2244	. . 3 |- (E!yE.xph -> E*yE.xph)
2		2moex 2275	. . 3 |- (E*yE.xph -> A.xE*yph)
3		2eu1 2284	. . . 4 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
4		simpl 443	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E!xE.yph)
5	3, 4	syl6bi 219	. . 3 |- (A.xE*yph -> (E!xE!yph -> E!xE.yph))
6	1, 2, 5	3syl 18	. 2 |- (E!yE.xph -> (E!xE!yph -> E!xE.yph))
7		2exeu 2281	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
8	7	expcom 424	. 2 |- (E!yE.xph -> (E!xE.yph -> E!xE!yph))
9	6, 8	impbid 183	1 |- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  E*wmo 2205

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:  2eu8  2291