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Theorem nfceqdf 2311
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 2240 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
41, 3nfbidf 1532 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
54albidv 1817 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
6 df-nfc 2301 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
7 df-nfc 2301 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
85, 6, 73bitr4g 222 1 (𝜑 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346   = wceq 1348  wnf 1453  wcel 2141  wnfc 2299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301
This theorem is referenced by:  nfopd  3782  dfnfc2  3814  nfimad  4962  nffvd  5508
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