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Theorem nabbib 3037
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.) Definitial form. (Revised by Wolf Lammen, 5-Mar-2025.)
Assertion
Ref Expression
nabbib ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

Proof of Theorem nabbib
StepHypRef Expression
1 df-ne 2933 . 2 ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ¬ {𝑥𝜑} = {𝑥𝜓})
2 exnal 1821 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
3 xor3 382 . . . . 5 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
43exbii 1842 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
52, 4bitr3i 277 . . 3 (¬ ∀𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6 abbib 2796 . . 3 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
75, 6xchnxbir 333 . 2 (¬ {𝑥𝜑} = {𝑥𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
81, 7bitri 275 1 ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1531   = wceq 1533  wex 1773  {cab 2701  wne 2932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-ne 2933
This theorem is referenced by:  suppvalbr  8145
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