Theorem nfopabd 5138

Description: Bound-variable hypothesis builder for class abstraction. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)

Hypotheses

Ref	Expression
nfopabd.1	⊢ Ⅎ𝑥𝜑
nfopabd.2	⊢ Ⅎ𝑦𝜑
nfopabd.4	⊢ (𝜑 → Ⅎ𝑧𝜓)

Assertion

Ref	Expression
nfopabd	⊢ (𝜑 → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem nfopabd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-opab 5133	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2		nfv 1918	. . 3 ⊢ Ⅎ𝑤𝜑
3		nfopabd.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfopabd.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
5		nfvd 1919	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩)
6		nfopabd.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓)
7	5, 6	nfand 1901	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
8	4, 7	nfexd 2327	. . . 4 ⊢ (𝜑 → Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
9	3, 8	nfexd 2327	. . 3 ⊢ (𝜑 → Ⅎ𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
10	2, 9	nfabdw 2929	. 2 ⊢ (𝜑 → Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
11	1, 10	nfcxfrd 2905	1 ⊢ (𝜑 → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783  Ⅎwnf 1787  {cab 2715  Ⅎwnfc 2886  ⟨cop 4564  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-opab 5133

This theorem is referenced by:  nfopab  5139  nfttrcld  33696