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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  con4bid  320  soisoi  7330  isomin  7339  naddel1  8679  alephdom  10087  nn0n0n1ge2b  12600  om2uzlt2i  14018  sadcaddlem  16550  isprm5  16801  pcdvdsb  16964  om2noseqlt2  28568  expgt0b  33290  oexpreposd  43200  tfsconcatb0  44188  cvgdvgrat  45140  hashnnltb  45849