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Theorem nfceqdf 2177
 Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfceqdf.1 𝑥𝜑
nfceqdf.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
nfceqdf (𝜑 → (𝑥𝐴𝑥𝐵))

Proof of Theorem nfceqdf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfceqdf.1 . . . 4 𝑥𝜑
2 nfceqdf.2 . . . . 5 (𝜑𝐴 = 𝐵)
32eleq2d 2107 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
41, 3nfbidf 1432 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
54albidv 1705 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
6 df-nfc 2167 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
7 df-nfc 2167 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
85, 6, 73bitr4g 212 1 (𝜑 → (𝑥𝐴𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165 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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167 This theorem is referenced by:  nfopd  3566  dfnfc2  3598  nfimad  4677  nffvd  5187
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