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Theorem nfopdALT 37829
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, 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 3697 . . . 4 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
43adantr 481 . . 3 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
5 abidnf 3697 . . . 4 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
65adantl 482 . . 3 ((𝑥𝐴𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
74, 6opeq12d 4880 . 2 ((𝑥𝐴𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩ = ⟨𝐴, 𝐵⟩)
8 nfaba1 2911 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
9 nfaba1 2911 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
108, 9nfop 4888 . 2 𝑥⟨{𝑧 ∣ ∀𝑥 𝑧𝐴}, {𝑧 ∣ ∀𝑥 𝑧𝐵}⟩
111, 2, 7, 10nfded2 37826 1 (𝜑𝑥𝐴, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  {cab 2709  wnfc 2883  cop 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634
This theorem is referenced by: (None)
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