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Theorem nfceqi 2955
Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-12 2176. (Revised by Wolf Lammen, 19-Jun-2023.)
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 2884 . . . 4 (𝑦𝐴𝑦𝐵)
32nfbii 1853 . . 3 (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵)
43albii 1821 . 2 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵)
5 df-nfc 2941 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
6 df-nfc 2941 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
74, 5, 63bitr4i 306 1 (𝑥𝐴𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wnf 1785  wcel 2112  wnfc 2939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941
This theorem is referenced by:  nfcxfr  2956  nfcxfrd  2957
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