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

Proof of Theorem elex2VD
StepHypRef Expression
1 idn1 39288 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 39336 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 eleq1a 2830 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
41, 2, 3e12 39449 . . . . 5 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
54in2 39328 . . . 4 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝑥𝐵)   )
65gen11 39339 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝑥𝐵)   )
7 elisset 3351 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
81, 7e1a 39350 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
9 exim 1906 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
106, 8, 9e11 39411 . 2 (   𝐴𝐵   ▶   𝑥 𝑥𝐵   )
1110in1 39285 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
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)
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