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Metamath Proof Explorer

Theorem mpbirand

Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021)

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

Proof

Step Hyp Ref Expression
1 mpbirand.1 ⊢ ( 𝜑 → 𝜒 )
2 mpbirand.2 ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
3 1 biantrurd ⊢ ( 𝜑 → ( 𝜃 ↔ ( 𝜒 ∧ 𝜃 ) ) )
4 2 3 bitr4d ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )