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Theorem axregndlem1 9409
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 2050 . 2 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
2 nfae 2314 . . 3 𝑥𝑥 𝑥 = 𝑧
3 nfae 2314 . . . . . 6 𝑧𝑥 𝑥 = 𝑧
4 elirrv 8489 . . . . . . . . 9 ¬ 𝑥𝑥
5 elequ1 1995 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑥𝑧𝑥))
64, 5mtbii 316 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑧𝑥)
76sps 2053 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧𝑥)
87pm2.21d 118 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (𝑧𝑥 → ¬ 𝑧𝑦))
93, 8alrimi 2080 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
109anim2i 592 . . . 4 ((𝑥𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
1110expcom 451 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
122, 11eximd 2083 . 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 384  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171
This theorem is referenced by:  axregndlem2  9410  axregnd  9411
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