Theorem dfral2 3089

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3090. (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 3084	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
3		ralnex 3064	. 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 3052  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-ral 3053  df-rex 3063

This theorem is referenced by:  rexnal  3090  imaeqsalvOLD  7320  boxcutc  8891  infssuni  9258  ac6n  10407  indstr  12841  trfil3  23844  nosepon  27645  noinfbnd1lem4  27706  cuteq1  27825  tglowdim2ln  28735  nmobndseqi  30867  stri  32345  hstri  32353  reprinfz1  34800  bnj1204  35188  onvf1odlem4  35322  fvineqsneq  37667  poimirlem1  37872  n0elqs  38583