Theorem dfral2 3083

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3084. (Revised by Wolf Lammen, 9-Dec-2019.)

Assertion

Ref	Expression
dfral2	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Proof of Theorem dfral2

Step	Hyp	Ref	Expression
1		notnotb 315	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	ralbii 3078	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
3		ralnex 3058	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wral 3047  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-ral 3048  df-rex 3057

This theorem is referenced by:  rexnal  3084  imaeqsalvOLD  7293  boxcutc  8860  infssuni  9225  ac6n  10371  indstr  12809  trfil3  23798  nosepon  27599  noinfbnd1lem4  27660  cuteq1  27773  tglowdim2ln  28624  nmobndseqi  30751  stri  32229  hstri  32237  reprinfz1  34627  bnj1204  35016  onvf1odlem4  35142  fvineqsneq  37446  poimirlem1  37661  n0elqs  38360