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Theorem nfopdALT 35987
Description: Deduction version of bound-variable hypothesis builder nfop 4811. This shows how the deduction version of a not-free theorem such as nfop 4811 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfopdALT.1 (𝜑𝑥𝐴)
nfopdALT.2 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfopdALT (𝜑𝑥𝐴, 𝐵⟩)

Proof of Theorem nfopdALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfopdALT.1 . 2 (𝜑𝑥𝐴)
2 nfopdALT.2 . 2 (𝜑𝑥𝐵)
3 abidnf 3691 . . . 4 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
43adantr 481 . . 3 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
5 abidnf 3691 . . . 4 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
65adantl 482 . . 3 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
74, 6opeq12d 4803 . 2 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
8 nfaba1 2983 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
9 nfaba1 2983 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
108, 9nfop 4811 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
111, 2, 7, 10nfded2 35984 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526   = wceq 1528  wcel 2105  {cab 2796  wnfc 2958  cop 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564
This theorem is referenced by: (None)
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