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Theorem rexm 3320
 Description: Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
Assertion
Ref Expression
rexm (∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexm
StepHypRef Expression
1 df-rex 2312 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpl 102 . . 3 ((𝑥𝐴𝜑) → 𝑥𝐴)
32eximi 1491 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
41, 3sylbi 114 1 (∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)
 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-ial 1427 This theorem depends on definitions:  df-bi 110  df-rex 2312 This theorem is referenced by:  eusvobj2  5498
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