Theorem bi1 117

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

Assertion

Ref	Expression
bi1	⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

Proof of Theorem bi1

Step	Hyp	Ref	Expression
1		df-bi 116	. . 3 ⊢ (((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2	1	simpli 110	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3	2	simpld 111	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  biimpi  119  bicom1  130  biimpd  143  ibd  177  pm5.74  178  bi3ant  223  pm5.501  243  pm5.32d  445  notbi  655  pm5.19  695  con4biddc  842  con1biimdc  858  bijadc  867  pclem6  1352  albi  1444  exbi  1583  equsexd  1707  cbv2h  1724  sbiedh  1760  eumo0  2030  ceqsalt  2712  vtoclgft  2736  spcgft  2763  pm13.183  2822  reu6  2873  reu3  2874  sbciegft  2939  ddifstab  3208  exmidsssnc  4126  fv3  5444  prnmaxl  7296  prnminu  7297  elabgft1  12985  elabgf2  12987  bj-axemptylem  13090  bj-inf2vn  13172  bj-inf2vn2  13173  bj-nn0sucALT  13176