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Theorem exanaliim 1535
 Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
exanaliim (x(φ ¬ ψ) → ¬ x(φψ))

Proof of Theorem exanaliim
StepHypRef Expression
1 annimim 781 . . 3 ((φ ¬ ψ) → ¬ (φψ))
21eximi 1488 . 2 (x(φ ¬ ψ) → x ¬ (φψ))
3 exnalim 1534 . 2 (x ¬ (φψ) → ¬ x(φψ))
42, 3syl 14 1 (x(φ ¬ ψ) → ¬ x(φψ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀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:  rexnalim  2311  nssr  2997  nssssr  3948  brprcneu  5112
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