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Theorem alexim 1518
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1492. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim (xφ → ¬ x ¬ φ)

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 539 . . . . 5 (φ → (¬ φ → ⊥ ))
21alimi 1324 . . . 4 (xφxφ → ⊥ ))
3 exim 1472 . . . 4 (xφ → ⊥ ) → (x ¬ φx ⊥ ))
42, 3syl 14 . . 3 (xφ → (x ¬ φx ⊥ ))
5 nfv 1402 . . . 4 x
6519.9 1517 . . 3 (x ⊥ ↔ ⊥ )
74, 6syl6ib 150 . 2 (xφ → (x ¬ φ → ⊥ ))
8 dfnot 1247 . 2 x ¬ φ ↔ (x ¬ φ → ⊥ ))
97, 8sylibr 137 1 (xφ → ¬ x ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1226  wfal 1233  wex 1362
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330
This theorem is referenced by:  exnalim  1519  exists2  1979
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