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Theorem rexnal3 3220
 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 3201 . . 3 (∃𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑧𝐶 𝜑)
212rexbii 3211 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑)
3 rexnal2 3219 . 2 (∃𝑥𝐴𝑦𝐵 ¬ ∀𝑧𝐶 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
42, 3bitri 278 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112 This theorem is referenced by:  tgdim01  26311
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