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Theorem axextnd 10005
Description: A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2383. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.)
Assertion
Ref Expression
axextnd 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)

Proof of Theorem axextnd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfnae 2449 . . . . . . . 8 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfnae 2449 . . . . . . . 8 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
31, 2nfan 1893 . . . . . . 7 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
4 nfcvf 3005 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
54adantr 483 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
65nfcrd 2967 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑦)
7 nfcvf 3005 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
87adantl 484 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
98nfcrd 2967 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑧)
106, 9nfbid 1896 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤𝑦𝑤𝑧))
11 elequ1 2114 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
12 elequ1 2114 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
1311, 12bibi12d 348 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤𝑦𝑤𝑧) ↔ (𝑥𝑦𝑥𝑧)))
1413a1i 11 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤𝑦𝑤𝑧) ↔ (𝑥𝑦𝑥𝑧))))
153, 10, 14cbvald 2421 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤𝑦𝑤𝑧) ↔ ∀𝑥(𝑥𝑦𝑥𝑧)))
16 axextg 2793 . . . . . 6 (∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦 = 𝑧)
1715, 16syl6bir 256 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧))
18 19.8a 2172 . . . . 5 (𝑦 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
1917, 18syl6 35 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥(𝑥𝑦𝑥𝑧) → ∃𝑥 𝑦 = 𝑧))
2019ex 415 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑥𝑦𝑥𝑧) → ∃𝑥 𝑦 = 𝑧)))
21 ax6e 2394 . . . . 5 𝑥 𝑥 = 𝑧
22 ax7 2016 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
2322aleximi 1825 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧))
2421, 23mpi 20 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧)
2524a1d 25 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥𝑦𝑥𝑧) → ∃𝑥 𝑦 = 𝑧))
26 ax6e 2394 . . . . 5 𝑥 𝑥 = 𝑦
27 ax7 2016 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
28 equcomi 2017 . . . . . . 7 (𝑧 = 𝑦𝑦 = 𝑧)
2927, 28syl6 35 . . . . . 6 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑦 = 𝑧))
3029aleximi 1825 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧))
3126, 30mpi 20 . . . 4 (∀𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
3231a1d 25 . . 3 (∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑥𝑦𝑥𝑧) → ∃𝑥 𝑦 = 𝑧))
3320, 25, 32pm2.61ii 185 . 2 (∀𝑥(𝑥𝑦𝑥𝑧) → ∃𝑥 𝑦 = 𝑧)
343319.35ri 1873 1 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528  wex 1773  wnfc 2959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-nfc 2961
This theorem is referenced by:  zfcndext  10027  axextprim  32915  axextdfeq  33030  axextndbi  33037
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