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Theorem nfopd 3782
Description: Deduction version of bound-variable hypothesis builder nfop 3781. This shows how the deduction version of a not-free theorem such as nfop 3781 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 2318 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2318 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfop 3781 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
4 nfopd.2 . . 3 (𝜑𝑥𝐴)
5 nfopd.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2315 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2315 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1558 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 2898 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109adantr 274 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
11 abidnf 2898 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211adantl 275 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1310, 12opeq12d 3773 . . . 4 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
148, 13nfceqdf 2311 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
154, 5, 14syl2anc 409 . 2 (𝜑 → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
163, 15mpbii 147 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wcel 2141  {cab 2156  wnfc 2299  cop 3586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592
This theorem is referenced by:  nfbrd  4034  nfovd  5882
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