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Theorem rexex 2411
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 2355 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 108 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1532 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 119 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1422  wcel 1434  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-rex 2355
This theorem is referenced by:  reu3  2783  rmo2i  2905  dffo5  5348  halfnq  6663  nsmallnq  6665  0npr  6735  genpml  6769  genpmu  6770  ltexprlemm  6852  ltexprlemloc  6859
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