Theorem rexn0OLD 4410

Description: Obsolete version of rexn0 4406 as of 2-Sep-2024. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rexn0OLD	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexn0OLD

Step	Hyp	Ref	Expression
1		ne0i 4235	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	a1d 25	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝐴 ≠ ∅))
3	2	rexlimiv 3204	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  ∅c0 4227

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-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-dif 3863  df-nul 4228

This theorem is referenced by: (None)