Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfded Structured version   Visualization version   GIF version

Theorem nfded 33055
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 3341. The last is assigned to the inference form (e.g. 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2755. (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 2753 . . . 4 𝑥𝑥𝐴
4 nfded.2 . . . 4 (𝑥𝐴𝐵 = 𝐶)
53, 4nfceqdf 2746 . . 3 (𝑥𝐴 → (𝑥𝐵𝑥𝐶))
62, 5syl 17 . 2 (𝜑 → (𝑥𝐵𝑥𝐶))
71, 6mpbii 221 1 (𝜑𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = 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:  nfunidALT  33058
  Copyright terms: Public domain W3C validator