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Theorem nfopd 4889
Description: Deduction version of bound-variable hypothesis builder nfop 4888. This shows how the deduction version of a not-free theorem such as nfop 4888 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 2912 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2912 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfop 4888 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
4 nfopd.2 . . 3 (𝜑𝑥𝐴)
5 nfopd.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2907 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2907 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1898 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 3707 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109adantr 480 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
11 abidnf 3707 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211adantl 481 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1310, 12opeq12d 4880 . . . 4 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
148, 13nfceqdf 2900 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
154, 5, 14syl2anc 584 . 2 (𝜑 → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
163, 15mpbii 233 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wnfc 2889  cop 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632
This theorem is referenced by:  nfbrd  5188  dfid3  5580  nfovd  7461
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