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Theorem ax9vsep 3937
Description: Derive a weakened version of ax-9 1467, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3932 and Extensionality ax-ext 2067. In intuitionistic logic a9evsep 3936 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 3936 . 2 𝑥 𝑥 = 𝑦
2 exalim 1434 . 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 1285   = wceq 1287  wex 1424
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-ial 1470  ax-ext 2067  ax-sep 3932
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293
This theorem is referenced by: (None)
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