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Theorem nfceqi 2171
 Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfceqi.1 A = B
Assertion
Ref Expression
nfceqi (xAxB)

Proof of Theorem nfceqi
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfceqi.1 . . . . 5 A = B
21eleq2i 2101 . . . 4 (y Ay B)
32nfbii 1359 . . 3 (Ⅎx y A ↔ Ⅎx y B)
43albii 1356 . 2 (yx y Ayx y B)
5 df-nfc 2164 . 2 (xAyx y A)
6 df-nfc 2164 . 2 (xByx y B)
74, 5, 63bitr4i 201 1 (xAxB)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164 This theorem is referenced by:  nfcxfr  2172  nfcxfrd  2173
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