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Theorem ralnex 2294
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
Assertion
Ref Expression
ralnex (x A ¬ φ ↔ ¬ x A φ)

Proof of Theorem ralnex
StepHypRef Expression
1 df-ral 2289 . 2 (x A ¬ φx(x A → ¬ φ))
2 alinexa 1476 . . 3 (x(x A → ¬ φ) ↔ ¬ x(x A φ))
3 df-rex 2290 . . 3 (x A φx(x A φ))
42, 3xchbinxr 595 . 2 (x(x A → ¬ φ) ↔ ¬ x A φ)
51, 4bitri 173 1 (x A ¬ φ ↔ ¬ x A φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1226  wex 1362   wcel 1374  wral 2284  wrex 2285
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-ie2 1364
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-ral 2289  df-rex 2290
This theorem is referenced by:  rexalim  2297  ralinexa  2329  nrex  2389  nrexdv  2390  uni0b  3579  iindif2m  3698
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