Theorem con4bid 306

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

Hypothesis

Ref	Expression
con4bid.1	⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Assertion

Ref	Expression
con4bid	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem con4bid

Step	Hyp	Ref	Expression
1		con4bid.1	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
2	1	biimprd 238	. . 3 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
3	2	con4d 114	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4	1	biimpd 219	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))
5	3, 4	impcon4bid 217	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196

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 197

This theorem is referenced by:  notbid  307  notbi  308  had0  1583  cbvexdva  2319  sbcne12  4019  ordsucuniel  7066  rankr1a  8737  ltaddsub  10540  leaddsub  10542  supxrbnd1  12189  supxrbnd2  12190  ioo0  12238  ico0  12259  ioc0  12260  icc0  12261  fllt  12647  rabssnn0fi  12825  elcls  20925  rusgrnumwwlks  26941  chrelat3  29358  sltrec  32049  wl-sb8et  33464  infxrbnd2  39898  oddprmne2  41949  nnolog2flm1  42709