MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfopd Structured version   Visualization version   GIF version

Theorem nfopd 4570
Description: Deduction version of bound-variable hypothesis builder nfop 4569. This shows how the deduction version of a not-free theorem such as nfop 4569 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
Hypotheses
Ref Expression
nfopd.2 (𝜑𝑥𝐴)
nfopd.3 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfopd (𝜑𝑥𝐴, 𝐵⟩)

Proof of Theorem nfopd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2908 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2908 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfop 4569 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
4 nfopd.2 . . 3 (𝜑𝑥𝐴)
5 nfopd.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2905 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2905 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1977 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 3516 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109adantr 472 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
11 abidnf 3516 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211adantl 473 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1310, 12opeq12d 4561 . . . 4 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
148, 13nfceqdf 2898 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
154, 5, 14syl2anc 696 . 2 (𝜑 → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
163, 15mpbii 223 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wnfc 2889  cop 4327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328
This theorem is referenced by:  nfbrd  4850  dfid3  5175  nfovd  6838
  Copyright terms: Public domain W3C validator