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Theorem nfded 33272
 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. (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3342. The last is assigned to the inference form (e.g. Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2756. (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 2754 . . . 4 𝑥𝑥𝐴
4 nfded.2 . . . 4 (𝑥𝐴𝐵 = 𝐶)
53, 4nfceqdf 2747 . . 3 (𝑥𝐴 → (𝑥𝐵𝑥𝐶))
62, 5syl 17 . 2 (𝜑 → (𝑥𝐵𝑥𝐶))
71, 6mpbii 222 1 (𝜑𝑥𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740 This theorem is referenced by:  nfunidALT  33275
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