This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )