Theorem dfral2 2994

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2995. (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 304	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
3		ralnex 2992	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
4	2, 3	bitri 264	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  rexnal  2995  boxcutc  7951  infssuni  8257  ac6n  9307  indstr  11756  trfil3  21692  tglowdim2ln  25546  nmobndseqi  27634  stri  29116  hstri  29124  reprinfz1  30700  bnj1204  31080  nosepon  31818  poimirlem1  33410  n0elqs  34098