Theorem biimp 118

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.)

Assertion

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

Proof of Theorem biimp

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	simpld 112	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biimpi  120  bicom1  131  biimpd  144  ibd  178  pm5.74  179  bi3ant  224  pm5.501  244  pm5.32d  450  notbi  670  pm5.19  711  con4biddc  862  con1biimdc  878  bijadc  887  pclem6  1416  albi  1514  exbi  1650  equsexd  1775  cbv2h  1794  cbv2w  1796  sbiedh  1833  eumo0  2108  ceqsalt  2827  vtoclgft  2852  spcgft  2881  pm13.183  2942  reu6  2993  reu3  2994  sbciegft  3060  ddifstab  3337  exmidsssnc  4291  fv3  5658  prnmaxl  7698  prnminu  7699  elabgft1  16310  elabgf2  16312  bj-axemptylem  16423  bj-inf2vn  16505  bj-inf2vn2  16506  bj-nn0sucALT  16509