Theorem dfral2 3116

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3117. (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 318	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	ralbii 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
3		ralnex 3091	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wral 3079  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-ral 3080  df-rex 3090

This theorem is referenced by:  rexnal  3117  rab0  4342  imaeqsalvOLD  7362  boxcutc  8935  infssuni  9299  ac6n  10464  indstr  12935  trfil3  24045  nosepon  27829  noinfbnd1lem4  27890  cuteq1  28010  tglowdim2ln  28925  nmobndseqi  31131  stri  32609  hstri  32617  reprinfz1  35009  bnj1204  35400  onvf1odlem4  35590  fvineqsneq  38058  poimirlem1  38272  n0elqs  38981