Theorem biimprcd 160

Description: Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Hypothesis

Ref	Expression
biimpcd.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
biimprcd	⊢ (𝜒 → (𝜑 → 𝜓))

Proof of Theorem biimprcd

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (𝜒 → 𝜒)
2		biimpcd.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	syl5ibrcom 157	1 ⊢ (𝜒 → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biimparc  299  pm5.32  453  oplem1  984  ax11i  1762  equsex  1776  eleq1a  2306  ceqsalg  2844  cgsexg  2851  cgsex2g  2852  cgsex4g  2853  ceqsex  2854  spc2egv  2909  spc3egv  2911  csbiebt  3180  dfiin2g  4026  sotricim  4446  ralxfrALT  4590  iunpw  4603  opelxp  4781  ssrel  4840  ssrel2  4842  ssrelrel  4852  iss  5086  funcnvuni  5427  fun11iun  5637  tfrlem8  6551  eroveu  6862  fundmen  7049  nneneq  7113  fidifsnen  7127  prarloclem5  7817  prarloc  7820  recexprlemss1l  7952  recexprlemss1u  7953  uzin  9890  indstr  9928  elfzmlbp  10470  swrdnd  11355  isclwwlknx  16428