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Theorem nabbi 2925
 Description: Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
nabbi (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})

Proof of Theorem nabbi
StepHypRef Expression
1 df-ne 2824 . 2 ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ¬ {𝑥𝜑} = {𝑥𝜓})
2 exnal 1794 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
3 xor3 371 . . . . 5 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
43exbii 1814 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
52, 4bitr3i 266 . . 3 (¬ ∀𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6 abbi 2766 . . 3 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
75, 6xchnxbi 321 . 2 (¬ {𝑥𝜑} = {𝑥𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
81, 7bitr2i 265 1 (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1521   = wceq 1523  ∃wex 1744  {cab 2637   ≠ wne 2823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ne 2824 This theorem is referenced by:  suppvalbr  7344
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