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  2344  cbvex2v  2346  cbvex  2404  cbvex2  2417  2ralorOLD  3232  rexcom  3290  cbvrexfw  3305  ceqsex  3530  ceqsexv  3532  gencbval  3543  ceqsralbv  3657  snnzb  4718  raldifsnb  4796  uni0b  4933  opab0  5559  tsna1  38151  ralopabb  43424