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Theorem nd3 10583
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
nd3 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)

Proof of Theorem nd3
StepHypRef Expression
1 elirrv 9590 . . . 4 ¬ 𝑥𝑥
2 elequ2 2121 . . . 4 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
31, 2mtbii 325 . . 3 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
43sps 2178 . 2 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
5 sp 2176 . 2 (∀𝑧 𝑥𝑦𝑥𝑦)
64, 5nsyl 140 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-pr 5427  ax-reg 9586
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-v 3476  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by:  nd4  10584  axrepnd  10588  axpowndlem3  10593  axinfnd  10600  axacndlem3  10603  axacnd  10606
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