Theorem elex2OLD 3489

Description: Obsolete version of elex2 2812 as of 30-Nov-2024. (Contributed by Alan Sare, 25-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elex2OLD	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2OLD

Step	Hyp	Ref	Expression
1		eleq1a 2830	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
2	1	alrimiv 1927	. 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3		elisset 2817	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
4		exim 1834	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵))
5	2, 3, 4	sylc 65	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810

This theorem is referenced by: (None)