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Theorem reximdva0 3915
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 3913 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 reximdva0.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝜓)
32ex 450 . . . . . 6 (𝜑 → (𝑥𝐴𝜓))
43ancld 575 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
54eximdv 1843 . . . 4 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
65imp 445 . . 3 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
71, 6sylan2b 492 . 2 ((𝜑𝐴 ≠ ∅) → ∃𝑥(𝑥𝐴𝜓))
8 df-rex 2914 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8sylibr 224 1 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wcel 1987  wne 2790  wrex 2909  c0 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2914  df-v 3192  df-dif 3563  df-nul 3898
This theorem is referenced by:  n0snor2el  4339  hashgt12el  13166  refun0  21258  cstucnd  22028  supxrnemnf  29419  kerunit  29650  elpaddn0  34605
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