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Theorem nfceqi 2315
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 2244 . . . 4 (𝑦𝐴𝑦𝐵)
32nfbii 1473 . . 3 (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵)
43albii 1470 . 2 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵)
5 df-nfc 2308 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
6 df-nfc 2308 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
74, 5, 63bitr4i 212 1 (𝑥𝐴𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308
This theorem is referenced by:  nfcxfr  2316  nfcxfrd  2317
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