Theorem nabbib 3063

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	⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

Proof of Theorem nabbib

Step	Hyp	Ref	Expression
1		df-ne 2959	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2		exnal 1857	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ¬ ∀𝑥(𝜑 ↔ 𝜓))
3		xor3 385	. . . . 5 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	3	exbii 1878	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
5	2, 4	bitr3i 280	. . 3 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6		abbib 2832	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
7	5, 6	xchnxbir 336	. 2 ⊢ (¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
8	1, 7	bitri 278	1 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809  {cab 2741   ≠ wne 2958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-ne 2959

This theorem is referenced by:  suppvalbr  8161