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  2307  ralbiim  3093  reu8  3704  dfss2  3932  soeq2  5568  fun11  6590  dffo3  7074  dffo3f  7078  isnsg2  19088  isarchi  33136  axextprim  35688  biimpexp  35704  axextndbi  35792  bj-nnfbit  36740  bj-nnfbid  36741  ifpidg  43480  ifp1bi  43491  ifpbibib  43499  rp-fakeanorass  43502  frege54cor0a  43852  aibandbiaiffaiffb  46895  aibandbiaiaiffb  46896  afv2orxorb  47229