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Theorem rexnalim 2360
 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 2355 . 2 (∃𝑥𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜑))
2 exanaliim 1579 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2354 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylnibr 635 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥𝐴 𝜑)
51, 4sylbi 119 1 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1283  ∃wex 1422   ∈ wcel 1434  ∀wral 2349  ∃wrex 2350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-ral 2354  df-rex 2355 This theorem is referenced by:  ralexim  2361  iundif2ss  3751  alzdvds  10399
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