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Theorem rexlimd 2430
 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 2390 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd.2 . . 3 𝑥𝜒
54r19.23 2424 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒))
63, 5sylib 127 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  rexlimdv  2432  ralxfrALT  4199  fvmptt  5262  ffnfv  5323  nneneq  6320  ac6sfi  6352  prarloclem3step  6594  prmuloc2  6665  caucvgprprlemaddq  6806  lbzbi  8551
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