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Theorem elex2OLD 3505
Description: Obsolete version of elex2 2818 as of 30-Nov-2024. (Contributed by Alan Sare, 25-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
elex2OLD (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2OLD
StepHypRef Expression
1 eleq1a 2836 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
21alrimiv 1927 . 2 (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝑥𝐵))
3 elisset 2823 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
4 exim 1834 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
52, 3, 4sylc 65 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wex 1779  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by: (None)
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