Theorem nan 843

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 456	. 2 ⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2		imnan 405	. . 3 ⊢ ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
3	2	imbi2i 339	. 2 ⊢ ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ∧ 𝜒)))
4	1, 3	bitr2i 279	1 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))

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:  pm4.15  846  somincom  6128  wemaplem2  9534  alephval3  10182  hauspwpwf1  24299  rtprmirr  27081  sticksstones22  43198  icccncfext  46866  stoweidlem34  47013  stirlinglem5  47057  fourierdlem42  47128  etransc  47262