Theorem bi2anan9r 844

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)

Hypotheses

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

Assertion

Ref	Expression
bi2anan9r	⊢ ((θ ∧ φ) → ((ψ ∧ τ) ↔ (χ ∧ η)))

Proof of Theorem bi2anan9r

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		bi2an9.2	. . 3 ⊢ (θ → (τ ↔ η))
3	1, 2	bi2anan9 843	. 2 ⊢ ((φ ∧ θ) → ((ψ ∧ τ) ↔ (χ ∧ η)))
4	3	ancoms 439	1 ⊢ ((θ ∧ φ) → ((ψ ∧ τ) ↔ (χ ∧ η)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is used by: (None)