Theorem impbi 211

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)

Assertion

Ref	Expression
impbi	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))

Proof of Theorem impbi

Step	Hyp	Ref	Expression
1		df-bi 210	. . 3 ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2		simprim 167	. . 3 ⊢ (¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) → (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
3	1, 2	ax-mp 5	. 2 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
4	3	expi 166	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  impbii  212  impbidd  213  impbid21d  214  dfbi1  216  impimprbi  842  bj-bisym  37440  bj-moeub  37741  eqsbc2VD  45807  orbi1rVD  45815  3impexpVD  45823  3impexpbicomVD  45824  imbi12VD  45840  sbcim2gVD  45842  sb5ALTVD  45880