Theorem impcon4bid 227

Description: A variation on impbid 212 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 115	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
4	1, 3	impbid 212	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  con4bid  317  soisoi  7283  isomin  7292  naddel1  8623  alephdom  10003  nn0n0n1ge2b  12506  om2uzlt2i  13913  sadcaddlem  16426  isprm5  16677  pcdvdsb  16840  om2noseqlt2  28292  expgt0b  32890  oexpreposd  42754  tfsconcatb0  43772  cvgdvgrat  44740