This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem equs4v

Description: Version of equs4 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993) (Revised by BJ, 31-May-2019)

		Ref	Expression
	Assertion	equs4v	$⊢ \forall x (x = y \to φ) \to \exists x (x = y \land φ)$

Proof

Step	Hyp	Ref	Expression
1		ax6ev	$⊢ \exists x x = y$
2		exintr	$⊢ \forall x (x = y \to φ) \to (\exists x x = y \to \exists x (x = y \land φ))$
3	1 2	mpi	$⊢ \forall x (x = y \to φ) \to \exists x (x = y \land φ)$