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	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	syldan.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
Assertion	syldan	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		syldan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		syldan.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
3		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
4	3 1 2	syl2anc	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )