Theorem dfuni2 3893

Description: Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
dfuni2	|- U.A = {x | E.y e. A x e. y}

Distinct variable group:   x,y,A

Proof of Theorem dfuni2

Step	Hyp	Ref	Expression
1		df-uni 3892	. 2 |- U.A = {x | E.y(x e. y /\ y e. A)}
2		exancom 1586	. . . 4 |- (E.y(x e. y /\ y e. A) <-> E.y(y e. A /\ x e. y))
3		df-rex 2620	. . . 4 |- (E.y e. A x e. y <-> E.y(y e. A /\ x e. y))
4	2, 3	bitr4i 243	. . 3 |- (E.y(x e. y /\ y e. A) <-> E.y e. A x e. y)
5	4	abbii 2465	. 2 |- {x | E.y(x e. y /\ y e. A)} = {x | E.y e. A x e. y}
6	1, 5	eqtri 2373	1 |- U.A = {x | E.y e. A x e. y}

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  U.cuni 3891

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rex 2620  df-uni 3892

This theorem is referenced by:  nfuni  3897  nfunid  3898  unieq  3900  uniiun  4019