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Theorem reximdva0m 3236
 Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
Hypothesis
Ref Expression
reximdva0m.1 ((𝜑𝑥𝐴) → 𝜓)
Assertion
Ref Expression
reximdva0m ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem reximdva0m
StepHypRef Expression
1 reximdva0m.1 . . . . . 6 ((𝜑𝑥𝐴) → 𝜓)
21ex 108 . . . . 5 (𝜑 → (𝑥𝐴𝜓))
32ancld 308 . . . 4 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
43eximdv 1760 . . 3 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
54imp 115 . 2 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
6 df-rex 2312 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
75, 6sylibr 137 1 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-rex 2312 This theorem is referenced by: (None)
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