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Theorem elex22 3369
Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
elex22 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elex22
StepHypRef Expression
1 eleq1a 2845 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
2 eleq1a 2845 . . . 4 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
31, 2anim12ii 604 . . 3 ((𝐴𝐵𝐴𝐶) → (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
43alrimiv 2007 . 2 ((𝐴𝐵𝐴𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
5 elisset 3367 . . 3 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
65adantr 466 . 2 ((𝐴𝐵𝐴𝐶) → ∃𝑥 𝑥 = 𝐴)
7 exim 1909 . 2 (∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)))
84, 6, 7sylc 65 1 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  en3lplem1VD  39600
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