Theorem syldan

Description: A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005) (Proof shortened by Wolf Lammen, 6-Apr-2013)

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

Proof

Step	Hyp	Ref	Expression
1		syldan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		syldan.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
3		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
4	3 1 2	syl2anc	$$⊢\left(\phi \wedge \psi \right)\to \theta$$