Theorem bi2anan9r 639

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 638	. 2 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
4	3	ancoms 458	1 ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ 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 207  df-an 396

This theorem is referenced by:  efrn2lp  5666  ltsosr  11134  seqf1olem2  14083  seqf1o  14084  pcval  16882  sltlpss  27945  uspgr2wlkeq  29664  satf0op  35382  fmlafvel  35390  fneval  36353  prtlem5  38861  prjspval  42613  rmydioph  43026  wepwsolem  43054  aomclem8  43073  sprsymrelfolem2  47480