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Theorem reximdai 2417
 Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
Hypotheses
Ref Expression
reximdai.1 𝑥𝜑
reximdai.2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
Assertion
Ref Expression
reximdai (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))

Proof of Theorem reximdai
StepHypRef Expression
1 reximdai.1 . . 3 𝑥𝜑
2 reximdai.2 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
31, 2ralrimi 2390 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexim 2413 . 2 (∀𝑥𝐴 (𝜓𝜒) → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
53, 4syl 14 1 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐴 𝜒))
 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 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  reximdvai  2419
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