Theorem mpbi2and

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	mpbi2and.1	⊢ ( 𝜑 → 𝜓 )
	mpbi2and.2	⊢ ( 𝜑 → 𝜒 )
	mpbi2and.3	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) )
Assertion	mpbi2and	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		mpbi2and.1	⊢ ( 𝜑 → 𝜓 )
2		mpbi2and.2	⊢ ( 𝜑 → 𝜒 )
3		mpbi2and.3	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) )
4	1 2	jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
5	4 3	mpbid	⊢ ( 𝜑 → 𝜃 )