Theorem dfral2 3114

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 3115. (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 3109	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
3		ralnex 3089	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wral 3077  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3078  df-rex 3088

This theorem is used by:  rexnal  3115  rab0  4335  boxcutc  8962  infssuni  9328  ac6n  10556  indstr  13036  trfil3  24200  nosepon  28015  noinfbnd1lem4  28076  cuteq1  28196  tglowdim2ln  29113  nmobndseqi  31374  stri  32852  hstri  32860  reprinfz1  35244  bnj1204  35635  onvf1odlem4  35868  fvineqsneq  38315  poimirlem1  38519  n0elqs  39244