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Theorem elex22 3516
 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 2906 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
2 eleq1a 2906 . . . 4 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
31, 2anim12ii 619 . . 3 ((𝐴𝐵𝐴𝐶) → (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
43alrimiv 1921 . 2 ((𝐴𝐵𝐴𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
5 elisset 3504 . . 3 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
65adantr 483 . 2 ((𝐴𝐵𝐴𝐶) → ∃𝑥 𝑥 = 𝐴)
7 exim 1827 . 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 398  ∀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 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-cleq 2812  df-clel 2891 This theorem is referenced by:  en3lplem1VD  41167
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