Theorem pm5.18 345

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		pm5.501 330	. . . 4 ⊢ (φ → (¬ ψ ↔ (φ ↔ ¬ ψ)))
2	1	con1bid 320	. . 3 ⊢ (φ → (¬ (φ ↔ ¬ ψ) ↔ ψ))
3		pm5.501 330	. . 3 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
4	2, 3	bitr2d 245	. 2 ⊢ (φ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ)))
5		nbn2 334	. . . 4 ⊢ (¬ φ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ)))
6	5	con1bid 320	. . 3 ⊢ (¬ φ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ))
7		nbn2 334	. . 3 ⊢ (¬ φ → (¬ ψ ↔ (φ ↔ ψ)))
8	6, 7	bitr2d 245	. 2 ⊢ (¬ φ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ)))
9	4, 8	pm2.61i 156	1 ⊢ ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  xor3  346  pm5.19  349  pm5.16  860  dfbi3  863  xorass  1308