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Theorem r19.27m 3533
Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Hypothesis
Ref Expression
r19.27m.1 𝑥𝜓
Assertion
Ref Expression
r19.27m (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.27m
StepHypRef Expression
1 r19.26 2616 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
2 r19.27m.1 . . . 4 𝑥𝜓
32r19.3rm 3526 . . 3 (∃𝑥 𝑥𝐴 → (𝜓 ↔ ∀𝑥𝐴 𝜓))
43anbi2d 464 . 2 (∃𝑥 𝑥𝐴 → ((∀𝑥𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓)))
51, 4bitr4id 199 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wnf 1471  wex 1503  wcel 2160  wral 2468
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2182  df-clel 2185  df-ral 2473
This theorem is referenced by:  r19.27mv  3534  raaanlem  3543
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