Theorem pclem6 1022

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 531	. . 3 ⊢ (𝜓 → (¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
2		nbbn 387	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) ↔ ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
3	1, 2	sylib 220	. 2 ⊢ (𝜓 → ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
4	3	con2i 141	1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  rru  3773  nalset  5220