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Metamath Proof Explorer

Theorem bitr4di

Description: A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993)

Ref Expression
Hypotheses bitr4di.1 ⊢ φ → ψ ↔ χ
bitr4di.2 ⊢ θ ↔ χ
Assertion bitr4di ⊢ φ → ψ ↔ θ

Proof

Step Hyp Ref Expression
1 bitr4di.1 ⊢ φ → ψ ↔ χ
2 bitr4di.2 ⊢ θ ↔ χ
3 2 bicomi ⊢ χ ↔ θ
4 1 3 bitrdi ⊢ φ → ψ ↔ θ