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Metamath Proof Explorer

Theorem dvelimv

Description: Similar to dvelim with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 25-Jul-2015) (Proof shortened by Wolf Lammen, 30-Apr-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	dvelimv.1	$⊢ z = y \to (φ \leftrightarrow ψ)$
	Assertion	dvelimv	$⊢ \neg \forall x x = y \to (ψ \to \forall x ψ)$

Proof

Step	Hyp	Ref	Expression
1		dvelimv.1	$⊢ z = y \to (φ \leftrightarrow ψ)$
2		ax-5	$⊢ φ \to \forall x φ$
3	2 1	dvelim	$⊢ \neg \forall x x = y \to (ψ \to \forall x ψ)$