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Theorem ax9vsep 4051
 Description: Derive a weakened version of ax-9 1511, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4046 and Extensionality ax-ext 2121. In intuitionistic logic a9evsep 4050 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 4050 . 2 𝑥 𝑥 = 𝑦
2 exalim 1478 . 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 1329   = wceq 1331  ∃wex 1468 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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337 This theorem is referenced by: (None)
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