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Theorem rexnal2 3125
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 3090 . . 3 (∃𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 𝜑)
21rexbii 3084 . 2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑)
3 rexnal 3090 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 274 1 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wral 3051  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-ral 3052  df-rex 3061
This theorem is referenced by:  rexnal3  3126  2nreu  4439  nf1const  7307  cat1  18112  isnsgrp  18709  nn0prpw  36046  smprngopr  37764  aks6d1c6lem3  41882  fimgmcyc  42222  clsk1independent  43748  ichnreuop  47078
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