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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 ((φψ) ↔ ¬ (φ ↔ ¬ ψ))

Proof of Theorem pm5.18
StepHypRef Expression
1 pm5.501 330 . . . 4 (φ → (¬ ψ ↔ (φ ↔ ¬ ψ)))
21con1bid 320 . . 3 (φ → (¬ (φ ↔ ¬ ψ) ↔ ψ))
3 pm5.501 330 . . 3 (φ → (ψ ↔ (φψ)))
42, 3bitr2d 245 . 2 (φ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
5 nbn2 334 . . . 4 φ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ)))
65con1bid 320 . . 3 φ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ))
7 nbn2 334 . . 3 φ → (¬ ψ ↔ (φψ)))
86, 7bitr2d 245 . 2 φ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
94, 8pm2.61i 156 1 ((φψ) ↔ ¬ (φ ↔ ¬ ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176 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 177 This theorem is referenced by:  xor3  346  pm5.19  349  pm5.16  860  dfbi3  863  xorass  1308
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