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Theorem r19.3rmv 3312
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
r19.3rmv (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem r19.3rmv
StepHypRef Expression
1 nfv 1421 . 2 𝑥𝜑
21r19.3rm 3310 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-ral 2311 This theorem is referenced by:  iinconstm  3666  cnvpom  4860  ssfiexmid  6336  diffitest  6344  caucvgsrlemasr  6874  resqrexlemgt0  9618
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