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	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfopdALT.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfopdALT	⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)

Proof of Theorem nfopdALT

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfopdALT.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		nfopdALT.2	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
3		abidnf 3697	. . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
4	3	adantr 481	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
5		abidnf 3697	. . . 4 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
6	5	adantl 482	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
7	4, 6	opeq12d 4880	. 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩)
8		nfaba1 2911	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
9		nfaba1 2911	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
10	8, 9	nfop 4888	. 2 ⊢ Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩
11	1, 2, 7, 10	nfded2 37826	1 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  Ⅎwnfc 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)