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Theorem nfcxfrdf 38948
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
Hypotheses
Ref Expression
nfcxfrdf.0 𝑥𝜑
nfcxfrdf.1 (𝜑𝐴 = 𝐵)
nfcxfrdf.2 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfcxfrdf (𝜑𝑥𝐴)

Proof of Theorem nfcxfrdf
StepHypRef Expression
1 nfcxfrdf.2 . 2 (𝜑𝑥𝐵)
2 nfcxfrdf.0 . . 3 𝑥𝜑
3 nfcxfrdf.1 . . 3 (𝜑𝐴 = 𝐵)
42, 3nfceqdf 2899 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
51, 4mpbird 257 1 (𝜑𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wnf 1780  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: (None)
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