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Theorem nfceqdf 2899
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2108 and df-clel 2814. (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 2731 . . . . 5 (𝐴 = 𝐵 → (𝑦𝐴𝑦𝐵))
42, 3syl 17 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
51, 4nfbidf 2222 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
65albidv 1918 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
7 df-nfc 2890 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
8 df-nfc 2890 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
96, 7, 83bitr4g 314 1 (𝜑 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-cleq 2727  df-nfc 2890
This theorem is referenced by:  nfopd  4895  dfnfc2  4934  nfimad  6089  nffvd  6919  riotasv2d  38939  nfcxfrdf  38948  nfded  38949  nfded2  38950  nfunidALT2  38951
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