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Theorem rexlimd 2546
Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
rexlimd.1 𝑥𝜑
rexlimd.2 𝑥𝜒
rexlimd.3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
Assertion
Ref Expression
rexlimd (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Proof of Theorem rexlimd
StepHypRef Expression
1 rexlimd.1 . . 3 𝑥𝜑
2 rexlimd.3 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
31, 2ralrimi 2503 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd.2 . . 3 𝑥𝜒
54r19.23 2540 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒))
63, 5sylib 121 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1436  wcel 1480  wral 2416  wrex 2417
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  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421  df-rex 2422
This theorem is referenced by:  rexlimdv  2548  ralxfrALT  4388  fvmptt  5512  ffnfv  5578  nneneq  6751  ac6sfi  6792  prarloclem3step  7316  prmuloc2  7387  caucvgprprlemaddq  7528  axpre-suploclemres  7721  lbzbi  9420  divalglemeunn  11629  divalglemeuneg  11631  oddpwdclemdvds  11859  oddpwdclemndvds  11860  trirec0  13298
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