Theorem dvdemo1 4359

Description: Demonstration of a theorem (scheme) that requires (meta)variables x and y to be distinct, but no others. It bundles the theorem schemes E. x ( x = y -> x e. x ) and E. x ( x = y -> y e. x ). Compare dvdemo2 4360. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)

Assertion

Ref	Expression
dvdemo1	|- E. x ( x = y -> z e. x )

Distinct variable group:   x, y

Proof of Theorem dvdemo1

Step	Hyp	Ref	Expression
1		dtru 4350	. . 3 |- -. A. x x = y
2		exnal 1580	. . 3 |- ( E. x -. x = y <-> -. A. x x = y )
3	1, 2	mpbir 201	. 2 |- E. x -. x = y
4		pm2.21 102	. 2 |- ( -. x = y -> ( x = y -> z e. x ) )
5	3, 4	eximii 1584	1 |- E. x ( x = y -> z e. x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1546   E.wex 1547

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-nul 4298  ax-pow 4337

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551