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Theorem nfccdeq 3466
Description: Variation of nfcdeq 3465 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 2787 . . 3 𝑥 𝑧𝐴
3 equid 1985 . . . . 5 𝑧 = 𝑧
43cdeqth 3455 . . . 4 CondEq(𝑥 = 𝑦𝑧 = 𝑧)
5 nfccdeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
64, 5cdeqel 3464 . . 3 CondEq(𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
72, 6nfcdeq 3465 . 2 (𝑧𝐴𝑧𝐵)
87eqriv 2648 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  wnfc 2780  CondEqwcdeq 3451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-cleq 2644  df-clel 2647  df-nfc 2782  df-cdeq 3452
This theorem is referenced by: (None)
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