Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexnal2 Structured version   Visualization version   GIF version

Theorem rexnal2 3219
 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 3201 . . 3 (∃𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 𝜑)
21rexbii 3210 . 2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑)
3 rexnal 3201 . 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 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:  rexnal3  3220  2nreu  4351  nf1const  7045  isnsgrp  17914  nn0prpw  33847  smprngopr  35557  clsk1independent  40836  ichnreuop  44073
 Copyright terms: Public domain W3C validator