Theorem nfopd 4848

Description: Deduction version of bound-variable hypothesis builder nfop 4847. This shows how the deduction version of a not-free theorem such as nfop 4847 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Hypotheses

Ref	Expression
nfopd.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfopd.3	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfopd	⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)

Proof of Theorem nfopd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2932	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2932	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfop 4847	. 2 ⊢ Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩
4		nfopd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfopd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2927	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2927	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1919	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3665	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	adantr 484	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
11		abidnf 3665	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	adantl 485	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
13	10, 12	opeq12d 4839	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩)
14	8, 13	nfceqdf 2920	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ ↔ Ⅎ𝑥⟨𝐴, 𝐵⟩))
15	4, 5, 14	syl2anc 593	. 2 ⊢ (𝜑 → (Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ ↔ Ⅎ𝑥⟨𝐴, 𝐵⟩))
16	3, 15	mpbii 235	1 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  {cab 2740  Ⅎwnfc 2909  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  nfbrd  5146  dfid3  5545  nfovd  7425