Theorem con4bii 321

Description: A contraposition inference. (Contributed by NM, 21-May-1994.)

Hypothesis

Ref	Expression
con4bii.1	⊢ (¬ 𝜑 ↔ ¬ 𝜓)

Assertion

Ref	Expression
con4bii	⊢ (𝜑 ↔ 𝜓)

Proof of Theorem con4bii

Step	Hyp	Ref	Expression
1		con4bii.1	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
2		notbi 319	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	mpbir 231	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ 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:  2false  375  equsexvw  2004  cbvexv1  2343  cbvex2v  2345  cbvex  2403  cbvex2  2416  2ralorOLD  3216  rexcom  3271  cbvrexfw  3285  ceqsex  3509  ceqsexv  3511  gencbval  3522  ceqsralbv  3636  snnzb  4694  raldifsnb  4772  uni0b  4909  opab0  5529  tsna1  38168  ralopabb  43435