Theorem impcon4bid 230

Description: A variation on impbid 215 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)

Hypotheses

Ref	Expression
impcon4bid.1	⊢ (𝜑 → (𝜓 → 𝜒))
impcon4bid.2	⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))

Assertion

Ref	Expression
impcon4bid	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem impcon4bid

Step	Hyp	Ref	Expression
1		impcon4bid.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		impcon4bid.2	. . 3 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))
3	2	con4d 116	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
4	1, 3	impbid 215	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

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:  con4bid  320  soisoi  7326  isomin  7335  naddel1  8670  alephdom  10061  nn0n0n1ge2b  12568  om2uzlt2i  13983  sadcaddlem  16510  isprm5  16761  pcdvdsb  16924  om2noseqlt2  28493  expgt0b  33161  oexpreposd  43103  tfsconcatb0  44091  cvgdvgrat  45043  hashnnltb  45752