Theorem zfun 4434

Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
zfun	|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )

Distinct variable group:   x, y, z

Proof of Theorem zfun

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-un 4433	. 2 |- E. x A. y ( E. w ( y e. w /\ w e. z ) -> y e. x )
2		elequ2 2153	. . . . . . 7 |- ( w = x -> ( y e. w <-> y e. x ) )
3		elequ1 2152	. . . . . . 7 |- ( w = x -> ( w e. z <-> x e. z ) )
4	2, 3	anbi12d 473	. . . . . 6 |- ( w = x -> ( ( y e. w /\ w e. z ) <-> ( y e. x /\ x e. z ) ) )
5	4	cbvexv 1918	. . . . 5 |- ( E. w ( y e. w /\ w e. z ) <-> E. x ( y e. x /\ x e. z ) )
6	5	imbi1i 238	. . . 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	albii 1470	. . 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	exbii 1605	. 2 |- ( E. x A. y ( E. w ( y e. w /\ w e. z ) -> y e. x ) <-> E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x ) )
9	1, 8	mpbi 145	1 |- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-14 2151  ax-un 4433

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  uniex2  4436  bj-uniex2  14550