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Theorem reximdva0 4350
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 4345 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 reximdva0.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝜓)
32ex 411 . . . . . 6 (𝜑 → (𝑥𝐴𝜓))
43ancld 549 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
54eximdv 1918 . . . 4 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
65imp 405 . . 3 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
71, 6sylan2b 592 . 2 ((𝜑𝐴 ≠ ∅) → ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3069 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8sylibr 233 1 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1779  wcel 2104  wne 2938  wrex 3068  c0 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-ne 2939  df-rex 3069  df-dif 3950  df-nul 4322
This theorem is referenced by:  n0snor2el  4833  f1cdmsn  7282  hashgt12el  14386  refun0  23239  cstucnd  24009  supxrnemnf  32248  kerunit  32707  ssmxidllem  32863  elpaddn0  38974  thinciso  47767
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