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Metamath Proof Explorer

Theorem impbid2

Description: Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007) (Proof shortened by Wolf Lammen, 27-Sep-2013)

Ref Expression
Hypotheses impbid2.1 ⊢ ( 𝜓 → 𝜒 )
impbid2.2 ⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
Assertion impbid2 ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 impbid2.1 ⊢ ( 𝜓 → 𝜒 )
2 impbid2.2 ⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
3 2 1 impbid1 ⊢ ( 𝜑 → ( 𝜒 ↔ 𝜓 ) )
4 3 bicomd ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )