Theorem con1bid 355

Description: A contraposition deduction. (Contributed by NM, 9-Oct-1999.)

Hypothesis

Ref	Expression
con1bid.1	⊢ (𝜑 → (¬ 𝜓 ↔ 𝜒))

Assertion

Ref	Expression
con1bid	⊢ (𝜑 → (¬ 𝜒 ↔ 𝜓))

Proof of Theorem con1bid

Step	Hyp	Ref	Expression
1		con1bid.1	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝜒))
2	1	bicomd 223	. . 3 ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))
3	2	con2bid 354	. 2 ⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒))
4	3	bicomd 223	1 ⊢ (𝜑 → (¬ 𝜒 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  pm5.18  381  necon1bbid  2978  r19.9rzv  4506  rexsng  4681  onmindif  6478  iotanul  6541  ondif2  8539  cnpart  15276  sadadd2lem2  16484  isnirred  20437  isreg2  23401  kqcldsat  23757  trufil  23934  itg2cnlem2  25812  issqf  27194  eupth2lem3lem4  30260  pjnorm2  31756  atdmd  32427  atmd2  32429  dfrdg4  35933  dalawlem13  39866  sticksstones1  42128  aks6d1c6lem4  42155  orddif0suc  43258  infordmin  43522