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Theorem elex22VD 39824
Description: Virtual deduction proof of elex22 3404. (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 39549 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝐴𝐵𝐴𝐶)   )
2 simpl 475 . . . . 5 ((𝐴𝐵𝐴𝐶) → 𝐴𝐵)
31, 2e1a 39611 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐵   )
4 elisset 3402 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
53, 4e1a 39611 . . 3 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥 𝑥 = 𝐴   )
6 idn2 39597 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq1a 2872 . . . . . . . 8 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
83, 6, 7e12 39709 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
9 simpr 478 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶) → 𝐴𝐶)
101, 9e1a 39611 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐶   )
11 eleq1a 2872 . . . . . . . 8 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
1210, 6, 11e12 39709 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐶   )
13 pm3.2 462 . . . . . . 7 (𝑥𝐵 → (𝑥𝐶 → (𝑥𝐵𝑥𝐶)))
148, 12, 13e22 39655 . . . . . 6 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   (𝑥𝐵𝑥𝐶)   )
1514in2 39589 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
1615gen11 39600 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
17 exim 1929 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)))
1816, 17e1a 39611 . . 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 39672 . 2 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥𝐵𝑥𝐶)   )
2120in1 39546 1 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651   = wceq 1653  wex 1875  wcel 2157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-v 3386  df-vd1 39545  df-vd2 39553
This theorem is referenced by: (None)
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