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Theorem r19.28mv 3455
 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 1508 . 2 𝑥𝜑
21r19.28m 3452 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∃wex 1468   ∈ wcel 1480  ∀wral 2416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135  df-ral 2421 This theorem is referenced by:  iinrabm  3875  iindif2m  3880  iinin2m  3881  xpiindim  4676  fintm  5308  ixpiinm  6618  neipsm  12337
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