Theorem jca

Description: Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Deduction form of pm3.2 and pm3.2i . Its associated deduction is jcad . Equivalent to the natural deduction rule /\ I ( /\ introduction), see natded . (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 25-Oct-2012)

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

Proof

Step	Hyp	Ref	Expression
1		jca.1	$$⊢\phi \to \psi$$
2		jca.2	$$⊢\phi \to \chi$$
3		pm3.2	$$⊢\psi \to \left(\chi \to \left(\psi \wedge \chi \right)\right)$$
4	1 2 3	sylc	$$⊢\phi \to \left(\psi \wedge \chi \right)$$