Theorem nfopd 4857

Description: Deduction version of bound-variable hypothesis builder nfop 4856. This shows how the deduction version of a not-free theorem such as nfop 4856 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 2935	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2935	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfop 4856	. 2 ⊢ Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩
4		nfopd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfopd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2930	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2930	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3667	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	adantr 486	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
11		abidnf 3667	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	adantl 487	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
13	10, 12	opeq12d 4848	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩)
14	8, 13	nfceqdf 2923	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ ↔ Ⅎ𝑥⟨𝐴, 𝐵⟩))
15	4, 5, 14	syl2anc 596	. 2 ⊢ (𝜑 → (Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ ↔ Ⅎ𝑥⟨𝐴, 𝐵⟩))
16	3, 15	mpbii 236	1 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912  ⟨cop 4597

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is used by:  nfbrd  5159  dfid3  5561  nfovd  7448