Theorem eu3 2230

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eu3.1	|- F/yph

Assertion

Ref	Expression
eu3	|- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 |- F/yph
2	1	eu2 2229	. 2 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
3	1	mo 2226	. . 3 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	anbi2i 675	. 2 |- ((E.xph /\ E.yA.x(ph -> x = y)) <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
5	2, 4	bitr4i 243	1 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544  [wsb 1648  E!weu 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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

This theorem is referenced by:  mo2  2233  eu5  2242  2eu4  2287  eqeu  3007  reu3  3026