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Metamath Proof Explorer

Theorem mpbiran2d

Description: Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022)

Ref Expression
Hypotheses mpbiran2d.1 ⊢ ( 𝜑 → 𝜃 )
mpbiran2d.2 ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
Assertion mpbiran2d ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 mpbiran2d.1 ⊢ ( 𝜑 → 𝜃 )
2 mpbiran2d.2 ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
3 2 biancomd ⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜃 ∧ 𝜒 ) ) )
4 1 3 mpbirand ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )