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  977  ax11i  1728  equsex  1742  eleq1a  2268  ceqsalg  2791  cgsexg  2798  cgsex2g  2799  cgsex4g  2800  ceqsex  2801  spc2egv  2854  spc3egv  2856  csbiebt  3124  dfiin2g  3950  sotricim  4359  ralxfrALT  4503  iunpw  4516  opelxp  4694  ssrel  4752  ssrel2  4754  ssrelrel  4764  iss  4993  funcnvuni  5328  fun11iun  5528  tfrlem8  6385  eroveu  6694  fundmen  6874  nneneq  6927  fidifsnen  6940  prarloclem5  7584  prarloc  7587  recexprlemss1l  7719  recexprlemss1u  7720  uzin  9651  indstr  9684  elfzmlbp  10224