Theorem nan 563

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 433	. 2 ⊢ (((φ ∧ ψ) → ¬ χ) ↔ (φ → (ψ → ¬ χ)))
2		imnan 411	. . 3 ⊢ ((ψ → ¬ χ) ↔ ¬ (ψ ∧ χ))
3	2	imbi2i 303	. 2 ⊢ ((φ → (ψ → ¬ χ)) ↔ (φ → ¬ (ψ ∧ χ)))
4	1, 3	bitr2i 241	1 ⊢ ((φ → ¬ (ψ ∧ χ)) ↔ ((φ ∧ ψ) → ¬ χ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  pm4.15  564