Theorem impbi 208

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 207	. . 3 ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2		simprim 166	. . 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 165	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

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

This theorem is referenced by:  impbii  209  impbidd  210  impbid21d  211  dfbi1  213  impimprbi  829  bj-bisym  36591  bj-moeub  36850  eqsbc2VD  44860  orbi1rVD  44868  3impexpVD  44876  3impexpbicomVD  44877  imbi12VD  44893  sbcim2gVD  44895  sb5ALTVD  44933