Theorem bi2anan9r 638

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 637	. 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  5654  ltsosr  11103  seqf1olem2  14025  seqf1o  14026  pcval  16798  sltlpss  27807  uspgr2wlkeq  29434  satf0op  34910  fmlafvel  34918  fneval  35759  prtlem5  38256  prjspval  41939  rmydioph  42347  wepwsolem  42378  aomclem8  42397  sprsymrelfolem2  46746