Theorem eueq 3009

Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eueq	|- (A e. _V <-> E!x x = A)

Distinct variable group:   x,A

Proof of Theorem eueq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2372	. . . 4 |- ((x = A /\ y = A) -> x = y)
2	1	gen2 1547	. . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
3	2	biantru 491	. 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
4		isset 2864	. 2 |- (A e. _V <-> E.x x = A)
5		eqeq1 2359	. . 3 |- (x = y -> (x = A <-> y = A))
6	5	eu4 2243	. 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
7	3, 4, 6	3bitr4i 268	1 |- (A e. _V <-> E!x x = A)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2860

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  ax-ext 2334

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  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  eueq1  3010  moeq  3013  fnopab2g  5207