Theorem mpbir2and

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011) (Proof shortened by Wolf Lammen, 24-Nov-2012)

	Ref	Expression
Hypotheses	mpbir2and.1	⊢ ( 𝜑 → 𝜒 )
	mpbir2and.2	⊢ ( 𝜑 → 𝜃 )
	mpbir2and.3	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
Assertion	mpbir2and	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		mpbir2and.1	⊢ ( 𝜑 → 𝜒 )
2		mpbir2and.2	⊢ ( 𝜑 → 𝜃 )
3		mpbir2and.3	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
4	1 2	jca	⊢ ( 𝜑 → ( 𝜒 ∧ 𝜃 ) )
5	4 3	mpbird	⊢ ( 𝜑 → 𝜓 )