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Theorem ax9vsep 3921
Description: Derive a weakened version of ax-9 1465, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3916 and Extensionality ax-ext 2065. In intuitionistic logic a9evsep 3920 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 3920 . 2 𝑥 𝑥 = 𝑦
2 exalim 1432 . 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 1283   = wceq 1285  wex 1422
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-ext 2065  ax-sep 3916
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by: (None)
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