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Theorem ralnex3 3249
 Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof shortened by Wolf Lammen, 18-May-2023.)
Assertion
Ref Expression
ralnex3 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑)

Proof of Theorem ralnex3
StepHypRef Expression
1 ralnex 3224 . . 3 (∀𝑧𝐶 ¬ 𝜑 ↔ ¬ ∃𝑧𝐶 𝜑)
212ralbii 3154 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 ¬ 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 ¬ ∃𝑧𝐶 𝜑)
3 ralnex2 3248 . 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 3126  ∃wrex 3127 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 3131  df-rex 3132 This theorem is referenced by:  axtgupdim2  26244
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