Theorem nan 827

Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)

Assertion

Ref	Expression
nan	⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))

Proof of Theorem nan

Step	Hyp	Ref	Expression
1		impexp 450	. 2 ⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2		imnan 399	. . 3 ⊢ ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
3	2	imbi2i 336	. 2 ⊢ ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ∧ 𝜒)))
4	1, 3	bitr2i 276	1 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  pm4.15  830  somincom  6125  wemaplem2  9538  alephval3  10101  hauspwpwf1  23813  sticksstones22  41477  rtprmirr  41726  icccncfext  45088  stoweidlem34  45235  stirlinglem5  45279  fourierdlem42  45350  etransc  45484