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Theorem nbbn 372
Description: Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.)
Assertion
Ref Expression
nbbn ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))

Proof of Theorem nbbn
StepHypRef Expression
1 xor3 371 . 2 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
2 con2bi 342 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
3 bicom 212 . 2 ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
41, 2, 33bitrri 287 1 ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
Colors of variables: wff setvar class
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:  biass  373  pclem6  1009  xorass  1617  hadbi  1686  canth  6771  qextltlem  12226  onint1  32754  notbinot1  34191  notbinot2  34195
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