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Metamath Proof Explorer

Theorem impbida

Description: Deduce an equivalence from two implications. Variant of impbid . (Contributed by NM, 17-Feb-2007)

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

Proof

Step Hyp Ref Expression
1 impbida.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2 impbida.2 ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜓 )
3 1 ex ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4 2 ex ⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
5 3 4 impbid ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )