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Theorem nfccdeq 2910
 Description: Variation of nfcdeq 2909 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
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 2276 . . 3 𝑥 𝑧𝐴
3 equid 1678 . . . . 5 𝑧 = 𝑧
43cdeqth 2899 . . . 4 CondEq(𝑥 = 𝑦𝑧 = 𝑧)
5 nfccdeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
64, 5cdeqel 2908 . . 3 CondEq(𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
72, 6nfcdeq 2909 . 2 (𝑧𝐴𝑧𝐵)
87eqriv 2137 1 𝐴 = 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  Ⅎwnfc 2269  CondEqwcdeq 2895 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-cdeq 2896 This theorem is referenced by: (None)
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