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Theorem rexnal3 3040
 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 2992 . . 3 (∃𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑧𝐶 𝜑)
212rexbii 3038 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑)
3 rexnal2 3039 . 2 (∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
42, 3bitri 264 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wral 2909  ∃wrex 2910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-ral 2914  df-rex 2915 This theorem is referenced by:  ralnex3  3042
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