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Theorem dvelimc 2929
Description: Version of dvelim 2454 for classes. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
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 1801 . . 3 𝑥
2 nftru 1801 . . 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 2928 . 2 (⊤ → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
109mptru 1544 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535   = wceq 1537  wtru 1538  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890
This theorem is referenced by: (None)
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