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Theorem nfceqdf 2900
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2109 and df-clel 2815. (Revised by WL and SN, 23-Aug-2024.)
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)
3 eleq2w2 2732 . . . . 5 (𝐴 = 𝐵 → (𝑦𝐴𝑦𝐵))
42, 3syl 17 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
51, 4nfbidf 2223 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
65albidv 1919 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
7 df-nfc 2891 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
8 df-nfc 2891 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
96, 7, 83bitr4g 314 1 (𝜑 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537   = wceq 1539  wnf 1782  wcel 2107  wnfc 2889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-cleq 2728  df-nfc 2891
This theorem is referenced by:  nfopd  4889  dfnfc2  4928  nfimad  6086  nffvd  6917  riotasv2d  38959  nfcxfrdf  38968  nfded  38969  nfded2  38970  nfunidALT2  38971
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