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Theorem reximdva0 4251
 Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
Hypothesis
Ref Expression
reximdva0.1 ((𝜑𝑥𝐴) → 𝜓)
Assertion
Ref Expression
reximdva0 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem reximdva0
StepHypRef Expression
1 n0 4246 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 reximdva0.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝜓)
32ex 417 . . . . . 6 (𝜑 → (𝑥𝐴𝜓))
43ancld 555 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
54eximdv 1919 . . . 4 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
65imp 411 . . 3 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
71, 6sylan2b 597 . 2 ((𝜑𝐴 ≠ ∅) → ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3077 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8sylibr 237 1 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1782   ∈ wcel 2112   ≠ wne 2952  ∃wrex 3072  ∅c0 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-ne 2953  df-rex 3077  df-dif 3862  df-nul 4227 This theorem is referenced by:  n0snor2el  4722  hashgt12el  13826  refun0  22208  cstucnd  22978  supxrnemnf  30608  kerunit  31041  ssmxidllem  31155  elpaddn0  37369
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