Theorem bi2anan9r 637

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 636	. 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
4	3	ancoms 458	1 ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  efrn2lp  5659  ltsosr  11092  seqf1olem2  14013  seqf1o  14014  pcval  16782  sltlpss  27635  uspgr2wlkeq  29167  satf0op  34663  fmlafvel  34671  fneval  35541  prtlem5  38034  prjspval  41648  rmydioph  42056  wepwsolem  42087  aomclem8  42106  sprsymrelfolem2  46461