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Metamath Proof Explorer

Theorem anim12d

Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994) (Proof shortened by Wolf Lammen, 18-Dec-2013)

Ref Expression
Hypotheses anim12d.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
anim12d.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion anim12d ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 anim12d.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 anim12d.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3 idd ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜏 ) ) )
4 1 2 3 syl2and ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜏 ) ) )