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Theorem reximdva0 4309
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 4307 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 reximdva0.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝜓)
32ex 413 . . . . . 6 (𝜑 → (𝑥𝐴𝜓))
43ancld 551 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
54eximdv 1909 . . . 4 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
65imp 407 . . 3 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
71, 6sylan2b 593 . 2 ((𝜑𝐴 ≠ ∅) → ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3141 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8sylibr 235 1 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  wne 3013  wrex 3136  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rex 3141  df-dif 3936  df-nul 4289
This theorem is referenced by:  n0snor2el  4756  hashgt12el  13771  refun0  22051  cstucnd  22820  supxrnemnf  30419  kerunit  30823  elpaddn0  36816
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