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Theorem axregndlem1 9636
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 2199 . 2 (𝑥𝑦 → ∃𝑥 𝑥𝑦)
2 nfae 2458 . . 3 𝑥𝑥 𝑥 = 𝑧
3 nfae 2458 . . . . . 6 𝑧𝑥 𝑥 = 𝑧
4 elirrv 8668 . . . . . . . . 9 ¬ 𝑥𝑥
5 elequ1 2146 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑥𝑧𝑥))
64, 5mtbii 315 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑧𝑥)
76sps 2202 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧𝑥)
87pm2.21d 118 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (𝑧𝑥 → ¬ 𝑧𝑦))
93, 8alrimi 2229 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))
109anim2i 594 . . . 4 ((𝑥𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
1110expcom 450 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → (𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))
122, 11eximd 2232 . 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 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by:  axregndlem2  9637  axregnd  9638
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