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Theorem rexnal3 3191
Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Assertion
Ref Expression
rexnal3 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Proof of Theorem rexnal3
StepHypRef Expression
1 rexnal 3141 . . 3 (∃𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑧𝐶 𝜑)
212rexbii 3189 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑)
3 rexnal2 3190 . 2 (∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
42, 3bitri 266 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:  ralnex3  3193
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