Theorem dvdemo1 4635

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 4636. ("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 4594	. . 3 |- -. A. x x = y
2		exnal 1699	. . 3 |- ( E. x -. x = y <-> -. A. x x = y )
3	1, 2	mpbir 213	. 2 |- E. x -. x = y
4		pm2.21 112	. 2 |- ( -. x = y -> ( x = y -> z e. x ) )
5	3, 4	eximii 1709	1 |- E. x ( x = y -> z e. x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1442   E.wex 1663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-nul 4534  ax-pow 4581

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668

This theorem is referenced by: (None)