Theorem im2anan9

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996)

	Ref	Expression
Hypotheses	im2an9.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	im2an9.2	$$⊢\theta \to \left(\tau \to \eta \right)$$
Assertion	im2anan9	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \wedge \tau \right)\to \left(\chi \wedge \eta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		im2an9.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		im2an9.2	$$⊢\theta \to \left(\tau \to \eta \right)$$
3	1	adantrd	$$⊢\phi \to \left(\left(\psi \wedge \tau \right)\to \chi \right)$$
4	2	adantld	$$⊢\theta \to \left(\left(\psi \wedge \tau \right)\to \eta \right)$$
5	3 4	anim12ii	$$⊢\left(\phi \wedge \theta \right)\to \left(\left(\psi \wedge \tau \right)\to \left(\chi \wedge \eta \right)\right)$$