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Theorem rexlimd 2447
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 2407 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd.2 . . 3 𝑥𝜒
54r19.23 2441 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒))
63, 5sylib 131 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1365  wcel 1409  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  rexlimdv  2449  ralxfrALT  4227  fvmptt  5290  ffnfv  5351  nneneq  6351  ac6sfi  6383  prarloclem3step  6652  prmuloc2  6723  caucvgprprlemaddq  6864  lbzbi  8648  divalglemeunn  10233  divalglemeuneg  10235  oddpwdclemdvds  10258  oddpwdclemndvds  10259
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