Theorem elex2VD 45413

Description: Virtual deduction proof of elex2 2839. (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 45150	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 45189	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3		eleq1a 2857	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	1, 2, 3	e12 45299	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐵   )
5	4	in2 45181	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
6	5	gen11 45192	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
7		elisset 2844	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
8	1, 7	e1a 45203	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   )
9		exim 1854	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵))
10	6, 8, 9	e11 45264	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 ∈ 𝐵   )
11	10	in1 45147	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-vd1 45146  df-vd2 45154

This theorem is referenced by: (None)