Theorem pm5.17 1029

Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Assertion

Ref	Expression
pm5.17	⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.17

Step	Hyp	Ref	Expression
1		bicom 225	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))
2		dfbi2 480	. 2 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ ((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓)))
3		orcom 884	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4		df-or 862	. . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑))
5	3, 4	bitr2i 279	. . 3 ⊢ ((¬ 𝜓 → 𝜑) ↔ (𝜑 ∨ 𝜓))
6		imnan 405	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
7	5, 6	anbi12i 640	. 2 ⊢ (((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
8	1, 2, 7	3bitrri 301	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  nbi2  1033  sgnneg  15233  odd2np1  16491  ordtconnlem1  34538