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Theorem nfceqdfOLD 2895
Description: Obsolete version of nfceqdf 2894 as of 23-Aug-2024. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfceqdf.1 𝑥𝜑
nfceqdf.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
nfceqdfOLD (𝜑 → (𝑥𝐴𝑥𝐵))

Proof of Theorem nfceqdfOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfceqdf.1 . . . 4 𝑥𝜑
2 nfceqdf.2 . . . . 5 (𝜑𝐴 = 𝐵)
32eleq2d 2815 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
41, 3nfbidf 2213 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
54albidv 1916 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
6 df-nfc 2881 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
7 df-nfc 2881 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
85, 6, 73bitr4g 314 1 (𝜑 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532   = wceq 1534  wnf 1778  wcel 2099  wnfc 2879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881
This theorem is referenced by: (None)
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