Theorem dfbi2 474

Description: A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
dfbi2	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))

Proof of Theorem dfbi2

Step	Hyp	Ref	Expression
1		dfbi1 213	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
2		df-an 396	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
3	1, 2	bitr4i 278	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → 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:  dfbi  475  pm4.71  557  impimprbi  828  pm5.17  1013  xor  1016  dfbi3  1049  ifpdfbi  1070  albiim  1889  nfbid  1902  sbbi  2308  ralbiim  3100  reu8  3716  dfss2  3944  soeq2  5583  fun11  6610  dffo3  7092  dffo3f  7096  isnsg2  19139  isarchi  33180  axextprim  35718  biimpexp  35734  axextndbi  35822  bj-nnfbit  36770  bj-nnfbid  36771  ifpidg  43515  ifp1bi  43526  ifpbibib  43534  rp-fakeanorass  43537  frege54cor0a  43887  aibandbiaiffaiffb  46923  aibandbiaiaiffb  46924  afv2orxorb  47257