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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