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Metamath Proof Explorer

Theorem bibi1d

Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993)

Ref Expression
Hypothesis imbid.1 ⊢ φ → ψ ↔ χ
Assertion bibi1d ⊢ φ → ψ ↔ θ ↔ χ ↔ θ

Proof

Step Hyp Ref Expression
1 imbid.1 ⊢ φ → ψ ↔ χ
2 1 bibi2d ⊢ φ → θ ↔ ψ ↔ θ ↔ χ
3 bicom ⊢ ψ ↔ θ ↔ θ ↔ ψ
4 bicom ⊢ χ ↔ θ ↔ θ ↔ χ
5 2 3 4 3bitr4g ⊢ φ → ψ ↔ θ ↔ χ ↔ θ