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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 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2VD
StepHypRef Expression
1 idn1 39549 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 39597 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 eleq1a 2872 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
41, 2, 3e12 39709 . . . . 5 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
54in2 39589 . . . 4 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝑥𝐵)   )
65gen11 39600 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝑥𝐵)   )
7 elisset 3402 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
81, 7e1a 39611 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
9 exim 1929 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
106, 8, 9e11 39672 . 2 (   𝐴𝐵   ▶   𝑥 𝑥𝐵   )
1110in1 39546 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
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)
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