Theorem 3jca

Description: Join consequents with conjunction. (Contributed by NM, 9-Apr-1994)

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

Proof

Step	Hyp	Ref	Expression
1		3jca.1	⊢ ( 𝜑 → 𝜓 )
2		3jca.2	⊢ ( 𝜑 → 𝜒 )
3		3jca.3	⊢ ( 𝜑 → 𝜃 )
4	1 2 3	jca31	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
5		df-3an	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
6	4 5	sylibr	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )