Theorem jcad

Description: Deduction conjoining the consequents of two implications. Deduction form of jca and double deduction form of pm3.2 and pm3.2i . (Contributed by NM, 15-Jul-1993) (Proof shortened by Wolf Lammen, 23-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		jcad.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		jcad.2	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3		pm3.2	⊢ ( 𝜒 → ( 𝜃 → ( 𝜒 ∧ 𝜃 ) ) )
4	1 2 3	syl6c	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ) ) )