Theorem pm4.61 410

Description: Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.61	⊢ (¬ (𝜑 → 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem pm4.61

Step	Hyp	Ref	Expression
1		annim 409	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
2	1	bicomi 227	1 ⊢ (¬ (𝜑 → 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401

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  df-an 402

This theorem is used by:  pm4.65  411  npss  4069  difin  4225  2nreu  4409  isf32lem2  10354  cat1  18181  degenmgm2nfun  19044  nmo  32912  hashxpe  33227  bnj1253  35475  fphpd  43621  clsk1independent  44850  nabctnabc  47746  islindeps  49310