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  672  pm5.19  714  con4biddc  865  con1biimdc  881  bijadc  890  pclem6  1419  albi  1517  exbi  1653  equsexd  1777  cbv2h  1796  cbv2w  1798  sbiedh  1835  eumo0  2110  ceqsalt  2830  vtoclgft  2855  spcgft  2884  pm13.183  2945  reu6  2996  reu3  2997  sbciegft  3063  ddifstab  3341  exmidsssnc  4299  fv3  5671  prnmaxl  7751  prnminu  7752  elabgft1  16479  elabgf2  16481  bj-axemptylem  16591  bj-inf2vn  16673  bj-inf2vn2  16674  bj-nn0sucALT  16677