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Theorem nfopd 4351
Description: Deduction version of bound-variable hypothesis builder nfop 4350. This shows how the deduction version of a not-free theorem such as nfop 4350 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 2755 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2755 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfop 4350 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
4 nfopd.2 . . 3 (𝜑𝑥𝐴)
5 nfopd.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2753 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2753 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1815 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 3341 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109adantr 479 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
11 abidnf 3341 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211adantl 480 . . . . 5 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1310, 12opeq12d 4342 . . . 4 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
148, 13nfceqdf 2746 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
154, 5, 14syl2anc 690 . 2 (𝜑 → (𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ ↔ 𝑥𝐴, 𝐵⟩))
163, 15mpbii 221 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  {cab 2595  wnfc 2737  cop 4130
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-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131
This theorem is referenced by:  nfbrd  4622  dfid3  4944  nfovd  6552
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