ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexnalim GIF version

Theorem rexnalim 2466
Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexnalim (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)

Proof of Theorem rexnalim
StepHypRef Expression
1 df-rex 2461 . 2 (∃𝑥𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜑))
2 exanaliim 1647 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2460 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylnibr 677 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥𝐴 𝜑)
51, 4sylbi 121 1 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1351  wex 1492  wcel 2148  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by:  nnral  2467  ralexim  2469  iundif2ss  3949  ixp0  6725  omniwomnimkv  7159  alzdvds  11840  pc2dvds  12309  isnsgrp  12701  nninfsellemeq  14416
  Copyright terms: Public domain W3C validator