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Theorem elex2VD 39387
Description: Virtual deduction proof of elex2 3247. (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 39107 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 39155 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 eleq1a 2725 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
41, 2, 3e12 39268 . . . . 5 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
54in2 39147 . . . 4 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝑥𝐵)   )
65gen11 39158 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝑥𝐵)   )
7 elisset 3246 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
81, 7e1a 39169 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
9 exim 1801 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
106, 8, 9e11 39230 . 2 (   𝐴𝐵   ▶   𝑥 𝑥𝐵   )
1110in1 39104 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521   = wceq 1523  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-vd1 39103  df-vd2 39111
This theorem is referenced by: (None)
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