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Theorem axregndlem1 10599
Description: Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
axregndlem1 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))

Proof of Theorem axregndlem1
StepHypRef Expression
1 19.8a 2166 . 2 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
2 nfae 2426 . . 3 𝑥𝑥 𝑥 = 𝑧
3 nfae 2426 . . . . . 6 𝑧𝑥 𝑥 = 𝑧
4 elirrv 9593 . . . . . . . . 9 ¬ 𝑥𝑥
5 elequ1 2105 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑥𝑧𝑥))
64, 5mtbii 326 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑧𝑥)
76sps 2170 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧𝑥)
87pm2.21d 121 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (𝑧𝑥 → ¬ 𝑧𝑦))
93, 8alrimi 2198 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
109anim2i 616 . . . 4 ((𝑥𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
1110expcom 413 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
122, 11eximd 2201 . 2 (∀𝑥 𝑥 = 𝑧 → (∃𝑥 𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
131, 12syl5 34 1 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1531  wex 1773
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by:  axregndlem2  10600  axregnd  10601
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