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Theorem nfccdeq 3717
 Description: Variation of nfcdeq 3716 for classes. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2158. (Revised by Gino Giotto, 19-May-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfccdeq.1 𝑥𝐴
nfccdeq.2 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
nfccdeq 𝐴 = 𝐵
Distinct variable groups:   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem nfccdeq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfccdeq.1 . . . 4 𝑥𝐴
21nfcri 2943 . . 3 𝑥 𝑧𝐴
3 eqid 2798 . . . . 5 𝑧 = 𝑧
43cdeqth 3706 . . . 4 CondEq(𝑥 = 𝑦𝑧 = 𝑧)
5 nfccdeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
64, 5cdeqel 3715 . . 3 CondEq(𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
72, 6nfcdeq 3716 . 2 (𝑧𝐴𝑧𝐵)
87eqriv 2795 1 𝐴 = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  CondEqwcdeq 3702 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-13 2379  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-cleq 2791  df-clel 2870  df-nfc 2938  df-cdeq 3703 This theorem is referenced by: (None)
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