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Theorem ralnex 2310
 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 2305 . 2 (x A ¬ φx(x A → ¬ φ))
2 alinexa 1491 . . 3 (x(x A → ¬ φ) ↔ ¬ x(x A φ))
3 df-rex 2306 . . 3 (x A φx(x A φ))
42, 3xchbinxr 607 . 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 1240  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301 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-ie2 1380 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-ral 2305  df-rex 2306 This theorem is referenced by:  rexalim  2313  ralinexa  2345  nrex  2405  nrexdv  2406  uni0b  3596  iindif2m  3715  icc0r  8565
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