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Theorem rexnalim 2428
Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexnalim (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)

Proof of Theorem rexnalim
StepHypRef Expression
1 df-rex 2423 . 2 (∃𝑥𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜑))
2 exanaliim 1627 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2422 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylnibr 667 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥𝐴 𝜑)
51, 4sylbi 120 1 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1330  wex 1469  wcel 1481  wral 2417  wrex 2418
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-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by:  ralexim  2430  iundif2ss  3885  ixp0  6632  omniwomnimkv  7048  alzdvds  11586  nninfsellemeq  13383
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