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Theorem dvelimc 2772
Description: Version of dvelim 2324 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 1720 . . 3 𝑥
2 nftru 1720 . . 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 2771 . 2 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
109trud 1483 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472   = wceq 1474  wtru 1475  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by:  nfcvf  2773
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