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Theorem nabbib 3045
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 2941 . 2 ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ¬ {𝑥𝜑} = {𝑥𝜓})
2 exnal 1827 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
3 xor3 382 . . . . 5 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
43exbii 1848 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
52, 4bitr3i 277 . . 3 (¬ ∀𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6 abbib 2811 . . 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 206  wal 1538   = wceq 1540  wex 1779  {cab 2714  wne 2940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941
This theorem is referenced by:  suppvalbr  8189
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