Theorem nabbi 3046

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 2943	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
2		exnal 1830	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ¬ ∀𝑥(𝜑 ↔ 𝜓))
3		xor3 383	. . . . 5 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	3	exbii 1851	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
5	2, 4	bitr3i 276	. . 3 ⊢ (¬ ∀𝑥(𝜑 ↔ 𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6		abbi 2811	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
7	5, 6	xchnxbi 331	. 2 ⊢ (¬ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
8	1, 7	bitr2i 275	1 ⊢ (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓})

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783  {cab 2715   ≠ wne 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-ne 2943

This theorem is referenced by:  suppvalbr  7952