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Theorem rexnal2 3179
Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
rexnal2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rexnal2
StepHypRef Expression
1 rexnal 3160 . . 3 (∃𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 𝜑)
21rexbii 3170 . 2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑)
3 rexnal 3160 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 278 1 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wral 3061  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-ral 3066  df-rex 3067
This theorem is referenced by:  rexnal3  3180  2nreu  4356  nf1const  7114  cat1  17603  isnsgrp  18167  nn0prpw  34249  smprngopr  35947  clsk1independent  41333  ichnreuop  44597
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