Theorem bi2bian9 845

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

Hypotheses

Ref	Expression
bi2an9.1	⊢ (φ → (ψ ↔ χ))
bi2an9.2	⊢ (θ → (τ ↔ η))

Assertion

Ref	Expression
bi2bian9	⊢ ((φ ∧ θ) → ((ψ ↔ τ) ↔ (χ ↔ η)))

Proof of Theorem bi2bian9

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	adantr 451	. 2 ⊢ ((φ ∧ θ) → (ψ ↔ χ))
3		bi2an9.2	. . 3 ⊢ (θ → (τ ↔ η))
4	3	adantl 452	. 2 ⊢ ((φ ∧ θ) → (τ ↔ η))
5	2, 4	bibi12d 312	1 ⊢ ((φ ∧ θ) → ((ψ ↔ τ) ↔ (χ ↔ η)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by: (None)