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Theorem dvelimc 2243
Description: Version of dvelim 1936 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 1396 . . 3 𝑥
2 nftru 1396 . . 3 𝑧
3 dvelimc.1 . . . 4 𝑥𝐴
43a1i 9 . . 3 (⊤ → 𝑥𝐴)
5 dvelimc.2 . . . 4 𝑧𝐵
65a1i 9 . . 3 (⊤ → 𝑧𝐵)
7 dvelimc.3 . . . 4 (𝑧 = 𝑦𝐴 = 𝐵)
87a1i 9 . . 3 (⊤ → (𝑧 = 𝑦𝐴 = 𝐵))
91, 2, 4, 6, 8dvelimdc 2242 . 2 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
109trud 1294 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283   = wceq 1285  wtru 1286  wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212
This theorem is referenced by:  nfcvf  2244
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