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Theorem r19.28mv 3527
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
r19.28mv (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.28mv
StepHypRef Expression
1 nfv 1538 . 2 𝑥𝜑
21r19.28m 3524 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1502  wcel 2158  wral 2465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-cleq 2180  df-clel 2183  df-ral 2470
This theorem is referenced by:  iinrabm  3961  iindif2m  3966  iinin2m  3967  xpiindim  4776  fintm  5413  ixpiinm  6737  neipsm  13925
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