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Theorem r19.3rmv 3513
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 1528 . 2 𝑥𝜑
21r19.3rm 3511 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wex 1492  wcel 2148  wral 2455
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-ral 2460
This theorem is referenced by:  iinconstm  3895  exmidsssnc  4202  cnvpom  5170  ssfilem  6872  diffitest  6884  inffiexmid  6903  ctssexmid  7145  exmidonfinlem  7189  caucvgsrlemasr  7786  resqrexlemgt0  11022
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