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Theorem rexex 2523
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 2461 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 110 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1600 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 121 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1492  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-rex 2461
This theorem is referenced by:  reu3  2927  rmo2i  3053  dffo5  5660  halfnq  7388  nsmallnq  7390  0npr  7460  genpml  7494  genpmu  7495  ltexprlemm  7577  ltexprlemloc  7584  dedekindeulemlub  13731  dedekindicclemlub  13740
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