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Metamath Proof Explorer

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 )