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Theorem rexnalim 2366
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 2361 . 2 (∃𝑥𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜑))
2 exanaliim 1581 . . 3 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 2360 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3sylnibr 635 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥𝐴 𝜑)
51, 4sylbi 119 1 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1285  wex 1424  wcel 1436  wral 2355  wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-ral 2360  df-rex 2361
This theorem is referenced by:  ralexim  2368  iundif2ss  3772  alzdvds  10649  nninfsellemeq  11262
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