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Theorem nfded2 38679
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., ((𝑥𝐴𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4888) that starts from abidnf 3695. The last is assigned to the inference form (e.g., 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4887) whose hypotheses are satisfied using nfaba1 2900. (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 2895 . . . . 5 𝑥𝑥𝐴
5 nfnfc1 2895 . . . . 5 𝑥𝑥𝐵
64, 5nfan 1895 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
7 nfded2.3 . . . 4 ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)
86, 7nfceqdf 2887 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐶𝑥𝐷))
92, 3, 8syl2anc 582 . 2 (𝜑 → (𝑥𝐶𝑥𝐷))
101, 9mpbii 232 1 (𝜑𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wnfc 2876
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-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2718  df-nfc 2878
This theorem is referenced by:  nfopdALT  38682
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