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Theorem ralnex2 3027
 Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
ralnex2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem ralnex2
StepHypRef Expression
1 notnotb 303 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
2 notnotb 303 . . . 4 (𝜑 ↔ ¬ ¬ 𝜑)
322rexbii 3024 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑)
4 rexnal2 3025 . . 3 (∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
53, 4bitr2i 264 . 2 (¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜑)
61, 5xchbinx 323 1 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902 This theorem is referenced by:  r2exlem  3041  axtgupdim2  25170  fourierdlem42  39042  uhgrvd00  40750
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