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  1778  cbv2h  1797  cbv2w  1799  sbiedh  1836  eumo0  2111  ceqsalt  2840  vtoclgft  2865  spcgft  2894  pm13.183  2955  reu6  3006  reu3  3007  sbciegft  3073  ddifstab  3351  exmidsssnc  4316  fv3  5693  prnmaxl  7803  prnminu  7804  elabgft1  16550  elabgf2  16552  bj-axemptylem  16662  bj-inf2vn  16744  bj-inf2vn2  16745  bj-nn0sucALT  16748