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Theorem axregndlem1 9303
 Description: Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
Assertion
Ref Expression
axregndlem1 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))

Proof of Theorem axregndlem1
StepHypRef Expression
1 19.8a 2039 . 2 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
2 nfae 2304 . . 3 𝑥𝑥 𝑥 = 𝑧
3 nfae 2304 . . . . . 6 𝑧𝑥 𝑥 = 𝑧
4 elirrv 8387 . . . . . . . . 9 ¬ 𝑥𝑥
5 elequ1 1984 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑥𝑧𝑥))
64, 5mtbii 315 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑧𝑥)
76sps 2043 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧𝑥)
87pm2.21d 117 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (𝑧𝑥 → ¬ 𝑧𝑦))
93, 8alrimi 2069 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
109anim2i 591 . . . 4 ((𝑥𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
1110expcom 450 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
122, 11eximd 2072 . 2 (∀𝑥 𝑥 = 𝑧 → (∃𝑥 𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
131, 12syl5 33 1 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  axregndlem2  9304  axregnd  9305
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