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Theorem ralnex 2403
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
Assertion
Ref Expression
ralnex (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)

Proof of Theorem ralnex
StepHypRef Expression
1 df-ral 2398 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 alinexa 1567 . . 3 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥𝐴𝜑))
3 df-rex 2399 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
42, 3xchbinxr 657 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥𝐴 𝜑)
51, 4bitri 183 1 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wal 1314  wex 1453  wcel 1465  wral 2393  wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie2 1455
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-ral 2398  df-rex 2399
This theorem is referenced by:  rexalim  2407  ralinexa  2439  nrex  2501  nrexdv  2502  ralnex2  2548  uni0b  3731  iindif2m  3850  f0rn0  5287  supmoti  6848  fodjuomnilemdc  6984  ismkvnex  6997  suprnubex  8679  icc0r  9677  ioo0  10005  ico0  10007  ioc0  10008  prmind2  11728  sqrt2irr  11767
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