Theorem dfbi2 386

Description: A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)

Assertion

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

Proof of Theorem dfbi2

Step	Hyp	Ref	Expression
1		df-bi 116	. . 3 ⊢ (((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2	1	simpli 110	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3	1	simpri 112	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
4	2, 3	impbii 125	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.71  387  pm5.17dc  890  dcbi  921  orbididc  938  trubifal  1395  albiim  1464  hbbi  1528  hbbid  1555  nfbid  1568  spsbbi  1817  sbbi  1933  cleqh  2240  ralbiim  2569  reu8  2884  sseq2  3126  soeq2  4246  fun11  5198  dffo3  5575  bdbi  13195