Theorem pm5.18 371

Description: Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)

Assertion

Ref	Expression
pm5.18	⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.18

Step	Hyp	Ref	Expression
1		pm5.501 356	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
2	1	con1bid 345	. . 3 ⊢ (𝜑 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓))
3		pm5.501 356	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
4	2, 3	bitr2d 269	. 2 ⊢ (𝜑 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
5		nbn2 360	. . . 4 ⊢ (¬ 𝜑 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
6	5	con1bid 345	. . 3 ⊢ (¬ 𝜑 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓))
7		nbn2 360	. . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))
8	6, 7	bitr2d 269	. 2 ⊢ (¬ 𝜑 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
9	4, 8	pm2.61i 176	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 196

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 197

This theorem is referenced by:  xor3  372  pm5.19  375  pm5.16  934  dfbi3OLD  995  xorneg2  1474