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Theorem elex22VD 39569
Description: Virtual deduction proof of elex22 3353. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex22VD ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elex22VD
StepHypRef Expression
1 idn1 39288 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝐴𝐵𝐴𝐶)   )
2 simpl 474 . . . . 5 ((𝐴𝐵𝐴𝐶) → 𝐴𝐵)
31, 2e1a 39350 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐵   )
4 elisset 3351 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
53, 4e1a 39350 . . 3 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥 𝑥 = 𝐴   )
6 idn2 39336 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq1a 2830 . . . . . . . 8 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
83, 6, 7e12 39449 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
9 simpr 479 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶) → 𝐴𝐶)
101, 9e1a 39350 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐶   )
11 eleq1a 2830 . . . . . . . 8 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
1210, 6, 11e12 39449 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐶   )
13 pm3.2 462 . . . . . . 7 (𝑥𝐵 → (𝑥𝐶 → (𝑥𝐵𝑥𝐶)))
148, 12, 13e22 39394 . . . . . 6 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   (𝑥𝐵𝑥𝐶)   )
1514in2 39328 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
1615gen11 39339 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
17 exim 1906 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)))
1816, 17e1a 39350 . . 3 (   (𝐴𝐵𝐴𝐶)   ▶   (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶))   )
19 pm2.27 42 . . 3 (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)) → ∃𝑥(𝑥𝐵𝑥𝐶)))
205, 18, 19e11 39411 . 2 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥𝐵𝑥𝐶)   )
2120in1 39285 1 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  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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