ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfccdeq GIF version

Theorem nfccdeq 2814
Description: Variation of nfcdeq 2813 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 2214 . . 3 𝑥 𝑧𝐴
3 equid 1630 . . . . 5 𝑧 = 𝑧
43cdeqth 2803 . . . 4 CondEq(𝑥 = 𝑦𝑧 = 𝑧)
5 nfccdeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
64, 5cdeqel 2812 . . 3 CondEq(𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
72, 6nfcdeq 2813 . 2 (𝑧𝐴𝑧𝐵)
87eqriv 2079 1 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  wnfc 2207  CondEqwcdeq 2799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-cdeq 2800
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator