Theorem 2eu7 2290

Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Assertion

Ref	Expression
2eu7	|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		nfe1 1732	. . . 4 |- F/xE.xph
2	1	nfeu 2220	. . 3 |- F/xE!yE.xph
3	2	euan 2261	. 2 |- (E!x(E!yE.xph /\ E.yph) <-> (E!yE.xph /\ E!xE.yph))
4		ancom 437	. . . . 5 |- ((E.xph /\ E.yph) <-> (E.yph /\ E.xph))
5	4	eubii 2213	. . . 4 |- (E!y(E.xph /\ E.yph) <-> E!y(E.yph /\ E.xph))
6		nfe1 1732	. . . . 5 |- F/yE.yph
7	6	euan 2261	. . . 4 |- (E!y(E.yph /\ E.xph) <-> (E.yph /\ E!yE.xph))
8		ancom 437	. . . 4 |- ((E.yph /\ E!yE.xph) <-> (E!yE.xph /\ E.yph))
9	5, 7, 8	3bitri 262	. . 3 |- (E!y(E.xph /\ E.yph) <-> (E!yE.xph /\ E.yph))
10	9	eubii 2213	. 2 |- (E!xE!y(E.xph /\ E.yph) <-> E!x(E!yE.xph /\ E.yph))
11		ancom 437	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E!yE.xph /\ E!xE.yph))
12	3, 10, 11	3bitr4ri 269	1 |- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Syntax hints:   <-> wb 176   /\ wa 358  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:  2eu8  2291