Theorem in3an

Description: The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a is the non-virtual deduction form of in3an . (Contributed by Alan Sare, 25-Jun-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	in3an.1	$$⊢\left(\phi ,\psi ,\left(\chi \wedge \theta \right)\to \tau \right)$$
Assertion	in3an	$$⊢\left(\phi ,\psi ,\chi \to \left(\theta \to \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		in3an.1	$$⊢\left(\phi ,\psi ,\left(\chi \wedge \theta \right)\to \tau \right)$$
2	1	dfvd3i	$$⊢\phi \to \left(\psi \to \left(\left(\chi \wedge \theta \right)\to \tau \right)\right)$$
3	2	exp4a	$$⊢\phi \to \left(\psi \to \left(\chi \to \left(\theta \to \tau \right)\right)\right)$$
4	3	dfvd3ir	$$⊢\left(\phi ,\psi ,\chi \to \left(\theta \to \tau \right)\right)$$