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 ) )