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Theorem rexm 3466
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 2423 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpl 108 . . 3 ((𝑥𝐴𝜑) → 𝑥𝐴)
32eximi 1580 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
41, 3sylbi 120 1 (∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)
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-ial 1515
This theorem depends on definitions:  df-bi 116  df-rex 2423
This theorem is referenced by:  eusvobj2  5767  exmidomni  7021  fodjum  7025
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