Theorem biimpr 130

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Assertion

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

Proof of Theorem biimpr

Step	Hyp	Ref	Expression
1		df-bi 117	. . 3 ⊢ (((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2	1	simpli 111	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3	2	simprd 114	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bicom1  131  pm5.74  179  bi3ant  224  pm5.32d  450  notbi  670  nbn2  702  pm4.72  832  con4biddc  862  con1biimdc  878  bijadc  887  pclem6  1416  exbir  1479  simplbi2comg  1486  albi  1514  exbi  1650  equsexd  1775  cbv2h  1794  cbv2w  1796  sbiedh  1833  ceqsalt  2826  spcegft  2882  elab3gf  2953  euind  2990  reu6  2992  reuind  3008  sbciegft  3059  iota4  5298  fv3  5650  algcvgblem  12571  bj-inf2vnlem1  16333