Theorem nan 514

Description: Theorem to move a conjunct in and out of a negation.

Assertion

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

Proof of Theorem nan

Step	Hyp	Ref	Expression
1		impexp 276	. 2 ⊢ (((φ ∧ ψ) → ¬ χ) ↔ (φ → (ψ → ¬ χ)))
2		imnan 207	. . 3 ⊢ ((ψ → ¬ χ) ↔ ¬ (ψ ∧ χ))
3	2	imbi2i 160	. 2 ⊢ ((φ → (ψ → ¬ χ)) ↔ (φ → ¬ (ψ ∧ χ)))
4	1, 3	bitr2 152	1 ⊢ ((φ → ¬ (ψ ∧ χ)) ↔ ((φ ∧ ψ) → ¬ χ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  cfsuc 3709

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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