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Metamath Proof Explorer

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 )