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Theorem ralnex 2624
 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 2619 . 2 (x A ¬ φx(x A → ¬ φ))
2 alinexa 1578 . . 3 (x(x A → ¬ φ) ↔ ¬ x(x A φ))
3 df-rex 2620 . . 3 (x A φx(x A φ))
42, 3xchbinxr 302 . 2 (x(x A → ¬ φ) ↔ ¬ x A φ)
51, 4bitri 240 1 (x A ¬ φ ↔ ¬ x A φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2619  df-rex 2620 This theorem is referenced by:  dfrex2  2627  ralinexa  2659  nrex  2716  nrexdv  2717  r19.43  2766  rabeq0  3572  iindif2  4035  evenodddisj  4516  rexiunxp  4824
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