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Theorem reximddv3 41296
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 467 . 2 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
3 reximddv3.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
42, 3reximddv 3272 1 (𝜑 → ∃𝑥𝐴 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-ral 3140  df-rex 3141
This theorem is referenced by:  rnmptlb  41390  rnmptbddlem  41391  limclner  41808  climisp  41903  climrescn  41905  liminflbuz2  41972  liminflimsupxrre  41974  climxlim2lem  42002
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