Theorem pclem6 1023

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 529	. . 3 ⊢ (𝜓 → (¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
2		nbbn 385	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) ↔ ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
3	1, 2	sylib 217	. 2 ⊢ (𝜓 → ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
4	3	con2i 139	1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396

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 397

This theorem is referenced by:  rru  3714  nalset  5237