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  5622  ltsosr  11054  seqf1olem2  14014  seqf1o  14015  pcval  16822  sltlpss  27826  uspgr2wlkeq  29581  satf0op  35371  fmlafvel  35379  fneval  36347  prtlem5  38860  prjspval  42598  rmydioph  43010  wepwsolem  43038  aomclem8  43057  sprsymrelfolem2  47498  pgnbgreunbgrlem1  48107  pgnbgreunbgrlem4  48113