Theorem 3jca

Description: Join consequents with conjunction. (Contributed by NM, 9-Apr-1994)

	Ref	Expression
Hypotheses	3jca.1	$$⊢\phi \to \psi$$
	3jca.2	$$⊢\phi \to \chi$$
	3jca.3	$$⊢\phi \to \theta$$
Assertion	3jca	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		3jca.1	$$⊢\phi \to \psi$$
2		3jca.2	$$⊢\phi \to \chi$$
3		3jca.3	$$⊢\phi \to \theta$$
4	1 2 3	jca31	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\wedge \theta \right)$$
5		df-3an	$$⊢\left(\psi \wedge \chi \wedge \theta \right)\leftrightarrow \left(\left(\psi \wedge \chi \right)\wedge \theta \right)$$
6	4 5	sylibr	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$