Theorem 3jcad

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005)

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

Proof

Step	Hyp	Ref	Expression
1		3jcad.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		3jcad.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3		3jcad.3	⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
4	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
5	2	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
6	3	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
7	4 5 6	3jca	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) )
8	7	ex	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )