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Theorem elex2VD 41917
 Description: Virtual deduction proof of elex2 3432. (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 41653 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 41692 . . . . . 6 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3 eleq1a 2847 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
41, 2, 3e12 41803 . . . . 5 (   𝐴𝐵   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
54in2 41684 . . . 4 (   𝐴𝐵   ▶   (𝑥 = 𝐴𝑥𝐵)   )
65gen11 41695 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 = 𝐴𝑥𝐵)   )
7 elisset 2833 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
81, 7e1a 41706 . . 3 (   𝐴𝐵   ▶   𝑥 𝑥 = 𝐴   )
9 exim 1835 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
106, 8, 9e11 41767 . 2 (   𝐴𝐵   ▶   𝑥 𝑥𝐵   )
1110in1 41650 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-vd1 41649  df-vd2 41657 This theorem is referenced by: (None)
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