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Metamath Proof Explorer

Theorem anim12dan

Description: Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011)

Ref Expression
Hypotheses anim12dan.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
anim12dan.2 ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
Assertion anim12dan ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜃 ) ) → ( 𝜒 ∧ 𝜏 ) )

Proof

Step Hyp Ref Expression
1 anim12dan.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2 anim12dan.2 ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
3 1 ex ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4 2 ex ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
5 3 4 anim12d ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜏 ) ) )
6 5 imp ⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜃 ) ) → ( 𝜒 ∧ 𝜏 ) )