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Theorem elex2VD 41037
Description: Virtual deduction proof of elex2 3522. (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 40773 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 40812 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 eleq1a 2913 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
41, 2, 3e12 40923 . . . . 5 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
54in2 40804 . . . 4 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝑥𝐵)   )
65gen11 40815 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝑥𝐵)   )
7 elisset 3511 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
81, 7e1a 40826 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
9 exim 1827 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
106, 8, 9e11 40887 . 2 (   𝐴𝐵   ▶   𝑥 𝑥𝐵   )
1110in1 40770 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528   = wceq 1530  wex 1773  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2819  df-clel 2898  df-vd1 40769  df-vd2 40777
This theorem is referenced by: (None)
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