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Metamath Proof Explorer

Theorem 3anim123i

Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994)

Ref Expression
Hypotheses 3anim123i.1 ⊢ ( 𝜑 → 𝜓 )
3anim123i.2 ⊢ ( 𝜒 → 𝜃 )
3anim123i.3 ⊢ ( 𝜏 → 𝜂 )
Assertion 3anim123i ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )

Proof

Step Hyp Ref Expression
1 3anim123i.1 ⊢ ( 𝜑 → 𝜓 )
2 3anim123i.2 ⊢ ( 𝜒 → 𝜃 )
3 3anim123i.3 ⊢ ( 𝜏 → 𝜂 )
4 1 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜓 )
5 2 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜃 )
6 3 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )
7 4 5 6 3jca ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )