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Theorem nfded2 36257
 Description: A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4785) that starts from abidnf 3645. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4784) whose hypotheses are satisfied using nfaba1 2966. (Contributed by NM, 19-Nov-2020.)
Hypotheses
Ref Expression
nfded2.1 (𝜑𝑥𝐴)
nfded2.2 (𝜑𝑥𝐵)
nfded2.3 ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)
nfded2.4 𝑥𝐶
Assertion
Ref Expression
nfded2 (𝜑𝑥𝐷)

Proof of Theorem nfded2
StepHypRef Expression
1 nfded2.4 . 2 𝑥𝐶
2 nfded2.1 . . 3 (𝜑𝑥𝐴)
3 nfded2.2 . . 3 (𝜑𝑥𝐵)
4 nfnfc1 2961 . . . . 5 𝑥𝑥𝐴
5 nfnfc1 2961 . . . . 5 𝑥𝑥𝐵
64, 5nfan 1900 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
7 nfded2.3 . . . 4 ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)
86, 7nfceqdf 2954 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐶𝑥𝐷))
92, 3, 8syl2anc 587 . 2 (𝜑 → (𝑥𝐶𝑥𝐷))
101, 9mpbii 236 1 (𝜑𝑥𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnfc 2939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941 This theorem is referenced by:  nfopdALT  36260
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