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Theorem reximdva0m 3382
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 114 . . . . 5 (𝜑 → (𝑥𝐴𝜓))
32ancld 323 . . . 4 (𝜑 → (𝑥𝐴 → (𝑥𝐴𝜓)))
43eximdv 1853 . . 3 (𝜑 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝜓)))
54imp 123 . 2 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥(𝑥𝐴𝜓))
6 df-rex 2423 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
75, 6sylibr 133 1 ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1469  wcel 1481  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-rex 2423
This theorem is referenced by: (None)
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