Mathbox for ML < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exellimddv Structured version   Visualization version   GIF version

Theorem exellimddv 32369
 Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 32368 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 1842 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
4 exellim 32368 . 2 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜓)) → 𝜓)
51, 3, 4syl2anc 691 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  topdifinffinlem  32371
 Copyright terms: Public domain W3C validator