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Metamath Proof Explorer

Theorem mpbidi

Description: A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994)

Ref Expression
Hypotheses mpbidi.min ⊢ ( 𝜃 → ( 𝜑 → 𝜓 ) )
mpbidi.maj ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion mpbidi ( 𝜃 → ( 𝜑 → 𝜒 ) )

Proof

Step Hyp Ref Expression
1 mpbidi.min ⊢ ( 𝜃 → ( 𝜑 → 𝜓 ) )
2 mpbidi.maj ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3 2 biimpd ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4 1 3 sylcom ⊢ ( 𝜃 → ( 𝜑 → 𝜒 ) )