Theorem con34b 319

Description: A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.)

Assertion

Ref	Expression
con34b	⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con34b

Step	Hyp	Ref	Expression
1		con3 156	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2		con4 113	. 2 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓))
3	1, 2	impbii 212	1 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209

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 210

This theorem is referenced by:  mtt  368  pm4.14  807  dfbi3  1050  ifpdfbi  1071  r19.23v  3198  raldifsni  4708  dff14a  7082  weniso  7163  dfom2  7646  dfsup2  9060  wemapsolem  9166  pwfseqlem3  10274  indstr  12512  rpnnen2lem12  15786  algcvgblem  16134  isirred2  19719  isdomn2  20337  ist0-3  22242  mdegleb  24962  dchrelbas4  26124  toslublem  30969  tosglblem  30971  poimirlem25  35539  poimirlem30  35544  tsbi3  36030  isdomn3  40732  ntrneikb  41381  aacllem  46176