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Theorem elex22 3420
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 2828 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
2 eleq1a 2828 . . . 4 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
31, 2anim12ii 621 . . 3 ((𝐴𝐵𝐴𝐶) → (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
43alrimiv 1934 . 2 ((𝐴𝐵𝐴𝐶) → ∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)))
5 elisset 2814 . . 3 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
65adantr 484 . 2 ((𝐴𝐵𝐴𝐶) → ∃𝑥 𝑥 = 𝐴)
7 exim 1840 . 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 399  wal 1540   = wceq 1542  wex 1786  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811
This theorem is referenced by:  en3lplem1VD  42023
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