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Theorem rexnalim 2314
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 2309 . 2 (∃𝑥𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜑))
2 exanaliim 1538 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2308 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylnibr 602 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥𝐴 𝜑)
51, 4sylbi 114 1 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wal 1241  wex 1381  wcel 1393  wral 2303  wrex 2304
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-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-ral 2308  df-rex 2309
This theorem is referenced by:  ralexim  2315  iundif2ss  3718
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