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Theorem alexnim 1522
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 1520 . . 3 (y ¬ φ → ¬ yφ)
21alimi 1323 . 2 (xy ¬ φx ¬ yφ)
3 alnex 1368 . 2 (x ¬ yφ ↔ ¬ xyφ)
42, 3sylib 127 1 (xy ¬ φ → ¬ xyφ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1314  wex 1361
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 532  ax-in2 533  ax-5 1315  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-4 1382  ax-17 1401  ax-ial 1410
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1232  df-nf 1329
This theorem is referenced by:  nalset  3839
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