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	$$⊢\phi \to \forall x\phi$$
	dvelim.2	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	dvelim	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dvelim.1	$$⊢\phi \to \forall x\phi$$
2		dvelim.2	$$⊢z=y\to \left(\phi \leftrightarrow \psi \right)$$
3		ax-5	$$⊢\psi \to \forall z\psi$$
4	1 3 2	dvelimh	$$⊢\neg \forall xx=y\to \left(\psi \to \forall x\psi \right)$$