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Theorem reximddv3 41413
 Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
reximddv3.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
reximddv3.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
reximddv3 (𝜑 → ∃𝑥𝐴 𝜒)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reximddv3
StepHypRef Expression
1 reximddv3.1 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
21anasss 469 . 2 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
3 reximddv3.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
42, 3reximddv 3275 1 (𝜑 → ∃𝑥𝐴 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  ∃wrex 3139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-ral 3143  df-rex 3144 This theorem is referenced by:  rnmptlb  41507  rnmptbddlem  41508  limclner  41925  climisp  42020  climrescn  42022  liminflbuz2  42089  liminflimsupxrre  42091  climxlim2lem  42119
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