Theorem elex2VD 39823

Description: Virtual deduction proof of elex2 3403. (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 39549	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 39597	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3		eleq1a 2872	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	1, 2, 3	e12 39709	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐵   )
5	4	in2 39589	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
6	5	gen11 39600	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)   )
7		elisset 3402	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
8	1, 7	e1a 39611	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   )
9		exim 1929	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵))
10	6, 8, 9	e11 39672	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 ∈ 𝐵   )
11	10	in1 39546	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1651   = wceq 1653  ∃wex 1875   ∈ wcel 2157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-v 3386  df-vd1 39545  df-vd2 39553

This theorem is referenced by: (None)