Theorem nabbib 3038

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 2936	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2		exnal 1834	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ¬ ∀𝑥(𝜑 ↔ 𝜓))
3		xor3 383	. . . . 5 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	3	exbii 1855	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
5	2, 4	bitr3i 278	. . 3 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6		abbib 2809	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
7	5, 6	xchnxbir 334	. 2 ⊢ (¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
8	1, 7	bitri 276	1 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 207  ∀wal 1545   = wceq 1547  ∃wex 1786  {cab 2718   ≠ wne 2935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-ne 2936

This theorem is referenced by:  suppvalbr  8111