Theorem pclem6 896

Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)

Assertion

Ref	Expression
pclem6	⊢ ((φ ↔ (ψ ∧ ¬ φ)) → ¬ ψ)

Proof of Theorem pclem6

Step	Hyp	Ref	Expression
1		ibar 490	. . 3 ⊢ (ψ → (¬ φ ↔ (ψ ∧ ¬ φ)))
2		nbbn 347	. . 3 ⊢ ((¬ φ ↔ (ψ ∧ ¬ φ)) ↔ ¬ (φ ↔ (ψ ∧ ¬ φ)))
3	1, 2	sylib 188	. 2 ⊢ (ψ → ¬ (φ ↔ (ψ ∧ ¬ φ)))
4	3	con2i 112	1 ⊢ ((φ ↔ (ψ ∧ ¬ φ)) → ¬ ψ)

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: (None)