Theorem nabbi 3089

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.)

Assertion

Ref	Expression
nabbi	⊢ (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓})

Proof of Theorem nabbi

Step	Hyp	Ref	Expression
1		df-ne 2988	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2		exnal 1828	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ¬ ∀𝑥(𝜑 ↔ 𝜓))
3		xor3 387	. . . . 5 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	3	exbii 1849	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
5	2, 4	bitr3i 280	. . 3 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6		abbi 2865	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
7	5, 6	xchnxbi 335	. 2 ⊢ (¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
8	1, 7	bitr2i 279	1 ⊢ (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓})

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781  {cab 2776   ≠ wne 2987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-ne 2988

This theorem is referenced by:  suppvalbr  7817