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Theorem nfceqdf 2757
 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 2684 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
41, 3nfbidf 2090 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
54albidv 1846 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
6 df-nfc 2750 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
7 df-nfc 2750 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
85, 6, 73bitr4g 303 1 (𝜑 → (𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  Ⅎwnfc 2748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707  df-cleq 2614  df-clel 2617  df-nfc 2750 This theorem is referenced by:  nfceqi  2758  nfopd  4394  dfnfc2  4427  dfnfc2OLD  4428  nfimad  5444  nffvd  6167  riotasv2d  33762  nfcxfrdf  33772  nfded  33773  nfded2  33774  nfunidALT2  33775
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