Theorem elex2VD 39568

Description: Virtual deduction proof of elex2 3352. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elex2VD	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2VD

Step	Hyp	Ref	Expression
1		idn1 39288	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 39336	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3		eleq1a 2830	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	1, 2, 3	e12 39449	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐵   )
5	4	in2 39328	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
6	5	gen11 39339	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
7		elisset 3351	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
8	1, 7	e1a 39350	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   )
9		exim 1906	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵))
10	6, 8, 9	e11 39411	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 ∈ 𝐵   )
11	10	in1 39285	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1626   = wceq 1628  ∃wex 1849   ∈ wcel 2135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-12 2192  ax-ext 2736

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1631  df-ex 1850  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-v 3338  df-vd1 39284  df-vd2 39292

This theorem is referenced by: (None)