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	⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

Proof of Theorem nabbib

Step	Hyp	Ref	Expression
1		df-ne 2941	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2		exnal 1827	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ¬ ∀𝑥(𝜑 ↔ 𝜓))
3		xor3 382	. . . . 5 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	3	exbii 1848	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
5	2, 4	bitr3i 277	. . 3 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6		abbib 2811	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
7	5, 6	xchnxbir 333	. 2 ⊢ (¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
8	1, 7	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))

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