Theorem pm5.16 1011

Description: Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.)

Assertion

Ref	Expression
pm5.16	⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.16

Step	Hyp	Ref	Expression
1		pm5.18 383	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	biimpi 215	. 2 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
3		imnan 400	. 2 ⊢ (((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) ↔ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)))
4	2, 3	mpbi 229	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: (None)