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Theorem reximdva0 4076
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 4074 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 reximdva0.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝜓)
32ex 449 . . . . . 6 (𝜑 → (𝑥𝐴𝜓))
43ancld 577 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
54eximdv 1995 . . . 4 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
65imp 444 . . 3 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
71, 6sylan2b 493 . 2 ((𝜑𝐴 ≠ ∅) → ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3056 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
97, 8sylibr 224 1 ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1853  wcel 2139  wne 2932  wrex 3051  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by:  n0snor2el  4509  hashgt12el  13402  refun0  21520  cstucnd  22289  supxrnemnf  29843  kerunit  30132  elpaddn0  35589
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