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Metamath Proof Explorer

Theorem bitr4id

Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994)

Ref Expression
Hypotheses bitr4id.2 ⊢ ψ ↔ χ
bitr4id.1 ⊢ φ → θ ↔ χ
Assertion bitr4id ⊢ φ → ψ ↔ θ

Proof

Step Hyp Ref Expression
1 bitr4id.2 ⊢ ψ ↔ χ
2 bitr4id.1 ⊢ φ → θ ↔ χ
3 1 bicomi ⊢ χ ↔ ψ
4 2 3 bitr2di ⊢ φ → ψ ↔ θ