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Metamath Proof Explorer

Theorem eexinst01

Description: exinst01 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eexinst01.1	$⊢ \exists x ψ$
	eexinst01.2	$⊢ φ \to (ψ \to χ)$
	eexinst01.3	$⊢ φ \to \forall x φ$
	eexinst01.4	$⊢ χ \to \forall x χ$
Assertion	eexinst01	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		eexinst01.1	$⊢ \exists x ψ$
2		eexinst01.2	$⊢ φ \to (ψ \to χ)$
3		eexinst01.3	$⊢ φ \to \forall x φ$
4		eexinst01.4	$⊢ χ \to \forall x χ$
5	3 4 2	exlimdh	$⊢ φ \to (\exists x ψ \to χ)$
6	1 5	mpi	$⊢ φ \to χ$