Theorem dvdemo2 4903

Description: Demonstration of a theorem (scheme) that requires (meta)variables x and z to be distinct, but no others. It bundles the theorem schemes E. x ( x = x -> z e. x ) and E. x ( x = y -> y e. x ). Compare dvdemo1 4902. (Contributed by NM, 1-Dec-2006.)

Assertion

Ref	Expression
dvdemo2	|- E. x ( x = y -> z e. x )

Distinct variable group:   x, z

Proof of Theorem dvdemo2

Step	Hyp	Ref	Expression
1		el 4847	. 2 |- E. x z e. x
2		ax-1 6	. 2 |- ( z e. x -> ( x = y -> z e. x ) )
3	1, 2	eximii 1764	1 |- E. x ( x = y -> z e. x )

Syntax hints:   -> wi 4   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-11 2034  ax-12 2047  ax-13 2246  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)