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Theorem rexex 2383
Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
rexex (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)

Proof of Theorem rexex
StepHypRef Expression
1 df-rex 2327 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 107 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1505 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 118 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1395  wcel 1407  wrex 2322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-ial 1441
This theorem depends on definitions:  df-bi 114  df-rex 2327
This theorem is referenced by:  reu3  2751  rmo2i  2873  dffo5  5341  halfnq  6537  nsmallnq  6539  0npr  6609  genpml  6643  genpmu  6644  ltexprlemm  6726  ltexprlemloc  6733
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