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Theorem rexex 2422
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 2365 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 108 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1536 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 119 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1426  wcel 1438  wrex 2360
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-rex 2365
This theorem is referenced by:  reu3  2805  rmo2i  2929  dffo5  5448  halfnq  6968  nsmallnq  6970  0npr  7040  genpml  7074  genpmu  7075  ltexprlemm  7157  ltexprlemloc  7164
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