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Theorem ax9vsep 4110
Description: Derive a weakened version of ax-9 1524, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4105 and Extensionality ax-ext 2152. In intuitionistic logic a9evsep 4109 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax9vsep ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem ax9vsep
StepHypRef Expression
1 a9evsep 4109 . 2 𝑥 𝑥 = 𝑦
2 exalim 1495 . 2 (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
31, 2ax-mp 5 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wal 1346   = wceq 1348  wex 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-ext 2152  ax-sep 4105
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354
This theorem is referenced by: (None)
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