Metamath Proof Explorer

Theorem dvelim

Description: This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A. x x = y as an antecedent. ph normally has z free and can be read ph ( z ) , and ps substitutes y for z and can be read ph ( y ) . We do not require that x and y be distinct: if they are not, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with A. x A. z , conjoin them, and apply dvelimdf .

Other variants of this theorem are dvelimh (with no distinct variable restrictions) and dvelimhw (that avoids ax-13 ). Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 23-Nov-1994) (New usage is discouraged.)

Ref Expression
Hypotheses dvelim.1 φ x φ
dvelim.2 z = y φ ψ
Assertion dvelim ¬ x x = y ψ x ψ


Step Hyp Ref Expression
1 dvelim.1 φ x φ
2 dvelim.2 z = y φ ψ
3 ax-5 ψ z ψ
4 1 3 2 dvelimh ¬ x x = y ψ x ψ