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Metamath Proof Explorer

Theorem impbid21d

Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013)

Ref Expression
Hypotheses impbid21d.1 ⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
impbid21d.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
Assertion impbid21d ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 impbid21d.1 ⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
2 impbid21d.2 ⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
3 impbi ⊢ ( ( 𝜒 → 𝜃 ) → ( ( 𝜃 → 𝜒 ) → ( 𝜒 ↔ 𝜃 ) ) )
4 1 2 3 syl2imc ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )