Theorem nfopd 4853

Description: Deduction version of bound-variable hypothesis builder nfop 4852. This shows how the deduction version of a not-free theorem such as nfop 4852 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Hypotheses

Ref	Expression
nfopd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfopd.2	⊢ (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 2934	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2934	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfop 4852	. 2 ⊢ Ⅎ𝑥⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩
4		nfopd.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfopd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2929	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2929	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1933	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3663	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	adantr 486	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
11		abidnf 3663	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	adantl 487	. . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
13	10, 12	opeq12d 4851	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩)
14	8, 13	nfceqdf 2922	. . 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 1567   = wceq 1569   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2911  ⟨cop 4838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839

This theorem is used by:  nfbrd  5154  dfid3  5553  nfovd  7459