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 230	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 205

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 206

This theorem is referenced by:  2false  376  equsexvw  2009  cbvexv1  2339  cbvex2v  2341  cbvex  2399  cbvex2  2412  2ralorOLD  3230  rexcom  3288  cbvrexfw  3303  ceqsex  3524  ceqsexv  3526  gencbval  3538  ceqsralbv  3646  snnzb  4723  raldifsnb  4800  uni0b  4938  opab0  5555  tsna1  37060  ralopabb  42210