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Theorem dvelimc 2816
 Description: Version of dvelim 2368 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
dvelimc.1 𝑥𝐴
dvelimc.2 𝑧𝐵
dvelimc.3 (𝑧 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
dvelimc (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Proof of Theorem dvelimc
StepHypRef Expression
1 nftru 1770 . . 3 𝑥
2 nftru 1770 . . 3 𝑧
3 dvelimc.1 . . . 4 𝑥𝐴
43a1i 11 . . 3 (⊤ → 𝑥𝐴)
5 dvelimc.2 . . . 4 𝑧𝐵
65a1i 11 . . 3 (⊤ → 𝑧𝐵)
7 dvelimc.3 . . . 4 (𝑧 = 𝑦𝐴 = 𝐵)
87a1i 11 . . 3 (⊤ → (𝑧 = 𝑦𝐴 = 𝐵))
91, 2, 4, 6, 8dvelimdc 2815 . 2 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
109trud 1533 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521   = wceq 1523  ⊤wtru 1524  Ⅎwnfc 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782 This theorem is referenced by:  nfcvf  2817
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