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Theorem rexex 2477
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 2420 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 simpr 109 . . 3 ((𝑥𝐴𝜑) → 𝜑)
32eximi 1579 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
41, 3sylbi 120 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1468  wcel 1480  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2420
This theorem is referenced by:  reu3  2869  rmo2i  2994  dffo5  5562  halfnq  7212  nsmallnq  7214  0npr  7284  genpml  7318  genpmu  7319  ltexprlemm  7401  ltexprlemloc  7408  dedekindeulemlub  12756  dedekindicclemlub  12765
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