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Theorem elex22 2586
 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 2125 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
2 eleq1a 2125 . . . 4 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
31, 2anim12ii 329 . . 3 ((𝐴𝐵𝐴𝐶) → (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
43alrimiv 1770 . 2 ((𝐴𝐵𝐴𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
5 elisset 2585 . . 3 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
65adantr 265 . 2 ((𝐴𝐵𝐴𝐶) → ∃𝑥 𝑥 = 𝐴)
7 exim 1506 . 2 (∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)))
84, 6, 7sylc 60 1 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101  ∀wal 1257   = wceq 1259  ∃wex 1397   ∈ wcel 1409 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576 This theorem is referenced by: (None)
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