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Theorem ax9vsep 4148
Description: Derive a weakened version of ax-9 1542, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4143 and Extensionality ax-ext 2171. In intuitionistic logic a9evsep 4147 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 4147 . 2 𝑥 𝑥 = 𝑦
2 exalim 1513 . 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 1362   = wceq 1364  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-ext 2171  ax-sep 4143
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370
This theorem is referenced by: (None)
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