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Metamath Proof Explorer

Theorem bi2bian9

Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004)

Ref Expression
Hypotheses bi2an9.1 ⊢ φ → ψ ↔ χ
bi2an9.2 ⊢ θ → τ ↔ η
Assertion bi2bian9 ⊢ φ ∧ θ → ψ ↔ τ ↔ χ ↔ η

Proof

Step Hyp Ref Expression
1 bi2an9.1 ⊢ φ → ψ ↔ χ
2 bi2an9.2 ⊢ θ → τ ↔ η
3 1 adantr ⊢ φ ∧ θ → ψ ↔ χ
4 2 adantl ⊢ φ ∧ θ → τ ↔ η
5 3 4 bibi12d ⊢ φ ∧ θ → ψ ↔ τ ↔ χ ↔ η