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Theorem nfceqi 2278
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 2207 . . . 4 (𝑦𝐴𝑦𝐵)
32nfbii 1450 . . 3 (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵)
43albii 1447 . 2 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵)
5 df-nfc 2271 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
6 df-nfc 2271 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
74, 5, 63bitr4i 211 1 (𝑥𝐴𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1330   = wceq 1332  wnf 1437  wcel 1481  wnfc 2269
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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by:  nfcxfr  2279  nfcxfrd  2280
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