Theorem axun 1081

Description: Axiom of Union expressed with fewest number of different variables.

Assertion

Ref	Expression
axun	|- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)

Distinct variable group(s):   x,y,z

Proof of Theorem axun

Step	Hyp	Ref	Expression
1		ax-un 1076	. 2 |- E.xA.y(E.w(y e. w /\ w e. z) -> y e. x)
2		a14b 820	. . . . . . 7 |- (w = x -> (y e. w <-> y e. x))
3		a13b 819	. . . . . . 7 |- (w = x -> (w e. z <-> x e. z))
4	2, 3	anbi12d 476	. . . . . 6 |- (w = x -> ((y e. w /\ w e. z) <-> (y e. x /\ x e. z)))
5	4	cbvexv 973	. . . . 5 |- (E.w(y e. w /\ w e. z) <-> E.x(y e. x /\ x e. z))
6	5	imbi1i 161	. . . 4 |- ((E.w(y e. w /\ w e. z) -> y e. x) <-> (E.x(y e. x /\ x e. z) -> y e. x))
7	6	bial 695	. . 3 |- (A.y(E.w(y e. w /\ w e. z) -> y e. x) <-> A.y(E.x(y e. x /\ x e. z) -> y e. x))
8	7	biex 733	. 2 |- (E.xA.y(E.w(y e. w /\ w e. z) -> y e. x) <-> E.xA.y(E.x(y e. x /\ x e. z) -> y e. x))
9	1, 8	mpbi 164	1 |- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  uniex 1947  axunndlem1 3741

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-un 1076

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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