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Metamath Proof Explorer

Theorem exinst11

Description: Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in Margaris p. 79 and E E. in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	exinst11.1	$⊢ (φ \to \exists x ψ)$
	exinst11.2	$⊢ (φ, ψ \to χ)$
	exinst11.3	$⊢ φ \to \forall x φ$
	exinst11.4	$⊢ χ \to \forall x χ$
Assertion	exinst11	$⊢ (φ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		exinst11.1	$⊢ (φ \to \exists x ψ)$
2		exinst11.2	$⊢ (φ, ψ \to χ)$
3		exinst11.3	$⊢ φ \to \forall x φ$
4		exinst11.4	$⊢ χ \to \forall x χ$
5	1	in1	$⊢ φ \to \exists x ψ$
6	2	dfvd2i	$⊢ φ \to (ψ \to χ)$
7	5 6 3 4	eexinst11	$⊢ φ \to χ$
8	7	dfvd1ir	$⊢ (φ \to χ)$