Theorem jca3

Description: Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010)

	Ref	Expression
Hypotheses	jca3.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	jca3.2	⊢ ( 𝜃 → 𝜏 )
Assertion	jca3	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → ( 𝜒 ∧ 𝜏 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		jca3.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		jca3.2	⊢ ( 𝜃 → 𝜏 )
3	1	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
4	3	a1d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜃 → 𝜒 ) )
5	4 2	jca2	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜃 → ( 𝜒 ∧ 𝜏 ) ) )
6	5	ex	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → ( 𝜒 ∧ 𝜏 ) ) ) )