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Theorem rexex 2368
 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 2312 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 103 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1491 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 114 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
 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:  reu3  2731  rmo2i  2848  dffo5  5316  halfnq  6509  nsmallnq  6511  0npr  6581  genpml  6615  genpmu  6616  ltexprlemm  6698  ltexprlemloc  6705
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