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Theorem ax6vsep 4818
Description: Derive ax6v 1946 (a weakened version of ax-6 1945 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 4814 and Extensionality ax-ext 2631. See ax6 2287 for the derivation of ax-6 1945 from ax6v 1946. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6vsep ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem ax6vsep
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4814 . . 3 𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
2 id 22 . . . . . . . . 9 (𝑧 = 𝑧𝑧 = 𝑧)
32biantru 525 . . . . . . . 8 (𝑧𝑦 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
43bibi2i 326 . . . . . . 7 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))))
54biimpri 218 . . . . . 6 ((𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → (𝑧𝑥𝑧𝑦))
65alimi 1779 . . . . 5 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∀𝑧(𝑧𝑥𝑧𝑦))
7 ax-ext 2631 . . . . 5 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
86, 7syl 17 . . . 4 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → 𝑥 = 𝑦)
98eximi 1802 . . 3 (∃𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
101, 9ax-mp 5 . 2 𝑥 𝑥 = 𝑦
11 df-ex 1745 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
1210, 11mpbi 220 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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