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Theorem nfccdeq 3787
Description: Variation of nfcdeq 3786 for classes. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2155. (Revised by GG, 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 group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem nfccdeq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfccdeq.1 . . . 4 𝑥𝐴
21nfcri 2895 . . 3 𝑥 𝑧𝐴
3 eqid 2735 . . . . 5 𝑧 = 𝑧
43cdeqth 3776 . . . 4 CondEq(𝑥 = 𝑦𝑧 = 𝑧)
5 nfccdeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
64, 5cdeqel 3785 . . 3 CondEq(𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
72, 6nfcdeq 3786 . 2 (𝑧𝐴𝑧𝐵)
87eqriv 2732 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wnfc 2888  CondEqwcdeq 3772
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-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-cleq 2727  df-clel 2814  df-nfc 2890  df-cdeq 3773
This theorem is referenced by: (None)
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