Theorem rabn0OLD 4094

Description: Obsolete proof of rabn0 4093 as of 16-Jul-2021. (Contributed by NM, 29-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rabn0OLD	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)

Proof of Theorem rabn0OLD

Step	Hyp	Ref	Expression
1		abn0 4089	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-rab 3051	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	neeq1i 2988	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅)
4		df-rex 3048	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	1, 3, 4	3bitr4i 292	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 196   ∧ wa 383  ∃wex 1845   ∈ wcel 2131  {cab 2738   ≠ wne 2924  ∃wrex 3043  {crab 3046  ∅c0 4050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-nul 4051

This theorem is referenced by:  rabeq0OLD  4095