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Theorem nfded 35983
Description: A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (𝑥𝐴 {𝑦 ∣ ∀𝑥𝑦𝐴} = 𝐴)) that starts from abidnf 3691. The last is assigned to the inference form (e.g., 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2983. (Contributed by NM, 19-Nov-2020.)
Hypotheses
Ref Expression
nfded.1 (𝜑𝑥𝐴)
nfded.2 (𝑥𝐴𝐵 = 𝐶)
nfded.3 𝑥𝐵
Assertion
Ref Expression
nfded (𝜑𝑥𝐶)

Proof of Theorem nfded
StepHypRef Expression
1 nfded.3 . 2 𝑥𝐵
2 nfded.1 . . 3 (𝜑𝑥𝐴)
3 nfnfc1 2977 . . . 4 𝑥𝑥𝐴
4 nfded.2 . . . 4 (𝑥𝐴𝐵 = 𝐶)
53, 4nfceqdf 2969 . . 3 (𝑥𝐴 → (𝑥𝐵𝑥𝐶))
62, 5syl 17 . 2 (𝜑 → (𝑥𝐵𝑥𝐶))
71, 6mpbii 234 1 (𝜑𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890  df-nfc 2960
This theorem is referenced by:  nfunidALT  35986
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