Metamath Proof Explorer

Theorem dvdemo2

Description: Demonstration of a theorem that requires the setvar variables x and z to be disjoint (but without any other disjointness conditions, and in particular, none on y ).

That theorem bundles the theorems ( |- E. x ( x = y -> z e. x ) with x , y , z disjoint), often called its "principal instance", and the two "degenerate instances" ( |- E. x ( x = x -> z e. x ) with x , z disjoint) and ( |- E. x ( x = z -> z e. x ) with x , z disjoint).

Compare with dvdemo1 , which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See for details on the "disjoint variable" mechanism.

Note that dvdemo2 is partially bundled, in that the pairs of setvar variables x , y and y , z need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 1-Dec-2006) (Revised by BJ, 13-Jan-2024)

Ref Expression
Assertion dvdemo2 x x = y z x


Step Hyp Ref Expression
1 el x z x
2 ax-1 z x x = y z x
3 1 2 eximii x x = y z x