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Theorem nfded2 33056
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 4351) that starts from abidnf 3341. The last is assigned to the inference form (e.g. 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4350) whose hypotheses are satisfied using nfaba1 2755. (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 2753 . . . . 5 𝑥𝑥𝐴
5 nfnfc1 2753 . . . . 5 𝑥𝑥𝐵
64, 5nfan 1815 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
7 nfded2.3 . . . 4 ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)
86, 7nfceqdf 2746 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥𝐶𝑥𝐷))
92, 3, 8syl2anc 690 . 2 (𝜑 → (𝑥𝐶𝑥𝐷))
101, 9mpbii 221 1 (𝜑𝑥𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by:  nfopdALT  33059
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