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Theorem ralnex3 3193
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 306 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑)
2 notnotb 306 . . . . 5 (𝜑 ↔ ¬ ¬ 𝜑)
32rexbii 3188 . . . 4 (∃𝑧𝐶 𝜑 ↔ ∃𝑧𝐶 ¬ ¬ 𝜑)
432rexbii 3189 . . 3 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ ¬ 𝜑)
5 rexnal3 3191 . . 3 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑)
64, 5bitr2i 267 . 2 (¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
71, 6xchbinx 325 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wral 3055  wrex 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-ral 3060  df-rex 3061
This theorem is referenced by:  axtgupdim2  25675
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