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Theorem rexnal2 3199
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 3180 . . 3 (∃𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 𝜑)
21rexbii 3189 . 2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑)
3 rexnal 3180 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 267 1 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wral 3083  wrex 3084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-ral 3088  df-rex 3089
This theorem is referenced by:  rexnal3  3200  ralnex2OLD  3202  2nreu  4271  isnsgrp  17769  nn0prpw  33225  smprngopr  34805  clsk1independent  39793  ichnreuop  43032
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