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  7336  isomin  7345  naddel1  8697  alephdom  10160  nn0n0n1ge2b  12675  om2uzlt2i  14094  sadcaddlem  16627  isprm5  16883  pcdvdsb  17047  om2noseqlt2  28686  expgt0b  33408  oexpreposd  43379  tfsconcatb0  44345  cvgdvgrat  45296  hashnnltb  46012