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Theorem alexnim 1536
 Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
alexnim (xy ¬ φ → ¬ xyφ)

Proof of Theorem alexnim
StepHypRef Expression
1 exnalim 1534 . . 3 (y ¬ φ → ¬ yφ)
21alimi 1341 . 2 (xy ¬ φx ¬ yφ)
3 alnex 1385 . 2 (x ¬ yφ ↔ ¬ xyφ)
42, 3sylib 127 1 (xy ¬ φ → ¬ xyφ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240  ∃wex 1378 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347 This theorem is referenced by:  nalset  3877  bj-nalset  8945
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