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Theorem nbbn 388
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 387 . 2 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
2 con2bi 357 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
3 bicom 225 . 2 ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
41, 2, 33bitrri 301 1 ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209
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 210
This theorem is referenced by:  biass  389  pclem6  1023  xorass  1507  hadbi  1599  canth  7095  qextltlem  12583  onint1  33871  wl-df3xor2  34845  wl-3xornot1  34856  wl-2xor  34859  notbinot1  35475  notbinot2  35479
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