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Theorem exellimddv 33323
Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 33322 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)
Hypotheses
Ref Expression
exellimddv.1 (𝜑 → ∃𝑥 𝑥𝐴)
exellimddv.2 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
exellimddv (𝜑𝜓)
Distinct variable groups:   𝜑,𝑥   𝜓,𝑥
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exellimddv
StepHypRef Expression
1 exellimddv.1 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
2 exellimddv.2 . . 3 (𝜑 → (𝑥𝐴𝜓))
32alrimiv 1895 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
4 exellim 33322 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜓)) → 𝜓)
51, 3, 4syl2anc 694 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  topdifinffinlem  33325
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