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	$$⊢\left(\phi \to \exists x\psi \right)$$
	exinst11.2	$$⊢\left(\phi ,\psi \to \chi \right)$$
	exinst11.3	$$⊢\phi \to \forall x\phi$$
	exinst11.4	$$⊢\chi \to \forall x\chi$$
Assertion	exinst11	$$⊢\left(\phi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exinst11.1	$$⊢\left(\phi \to \exists x\psi \right)$$
2		exinst11.2	$$⊢\left(\phi ,\psi \to \chi \right)$$
3		exinst11.3	$$⊢\phi \to \forall x\phi$$
4		exinst11.4	$$⊢\chi \to \forall x\chi$$
5	1	in1	$$⊢\phi \to \exists x\psi$$
6	2	dfvd2i	$$⊢\phi \to \left(\psi \to \chi \right)$$
7	5 6 3 4	eexinst11	$$⊢\phi \to \chi$$
8	7	dfvd1ir	$$⊢\left(\phi \to \chi \right)$$