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Theorem ralnex2 3074
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 304 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
2 notnotb 304 . . . 4 (𝜑 ↔ ¬ ¬ 𝜑)
322rexbii 3071 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑)
4 rexnal2 3072 . . 3 (∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
53, 4bitr2i 265 . 2 (¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜑)
61, 5xchbinx 323 1 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946  df-rex 2947
This theorem is referenced by:  r2exlem  3088  axtgupdim2  25415  uhgrvd00  26486  fourierdlem42  40684
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