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Theorem ax9vsep 3991
Description: Derive a weakened version of ax-9 1479, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3986 and Extensionality ax-ext 2082. In intuitionistic logic a9evsep 3990 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 3990 . 2 𝑥 𝑥 = 𝑦
2 exalim 1446 . 2 (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
31, 2ax-mp 7 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wal 1297   = wceq 1299  wex 1436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482  ax-ext 2082  ax-sep 3986
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305
This theorem is referenced by: (None)
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