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Theorem rexlimd 2578
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 2535 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd.2 . . 3 𝑥𝜒
54r19.23 2572 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒))
63, 5sylib 121 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1447  wcel 2135  wral 2442  wrex 2443
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-ral 2447  df-rex 2448
This theorem is referenced by:  rexlimdv  2580  ralxfrALT  4442  fvmptt  5574  ffnfv  5640  nneneq  6817  ac6sfi  6858  prarloclem3step  7431  prmuloc2  7502  caucvgprprlemaddq  7643  axpre-suploclemres  7836  lbzbi  9548  divalglemeunn  11852  divalglemeuneg  11854  oddpwdclemdvds  12096  oddpwdclemndvds  12097  trirec0  13816
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