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Theorem ralnex3 3028
 Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Assertion
Ref Expression
ralnex3 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Proof of Theorem ralnex3
StepHypRef Expression
1 notnotb 303 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑)
2 notnotb 303 . . . . 5 (𝜑 ↔ ¬ ¬ 𝜑)
32rexbii 3023 . . . 4 (∃𝑧𝐶 𝜑 ↔ ∃𝑧𝐶 ¬ ¬ 𝜑)
432rexbii 3024 . . 3 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ ¬ 𝜑)
5 rexnal3 3026 . . 3 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑)
64, 5bitr2i 264 . 2 (¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
71, 6xchbinx 323 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 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:  axtgupdim2  25170
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