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  666  pm5.19  706  con4biddc  857  con1biimdc  873  bijadc  882  pclem6  1374  albi  1468  exbi  1604  equsexd  1729  cbv2h  1748  cbv2w  1750  sbiedh  1787  eumo0  2057  ceqsalt  2764  vtoclgft  2788  spcgft  2815  pm13.183  2876  reu6  2927  reu3  2928  sbciegft  2994  ddifstab  3268  exmidsssnc  4204  fv3  5539  prnmaxl  7487  prnminu  7488  elabgft1  14533  elabgf2  14535  bj-axemptylem  14647  bj-inf2vn  14729  bj-inf2vn2  14730  bj-nn0sucALT  14733