Theorem cbvopab 5176

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
cbvopab.1	⊢ Ⅎ𝑧𝜑
cbvopab.2	⊢ Ⅎ𝑤𝜑
cbvopab.3	⊢ Ⅎ𝑥𝜓
cbvopab.4	⊢ Ⅎ𝑦𝜓
cbvopab.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopab	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem cbvopab

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1946	. . . . 5 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvopab.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1931	. . . 4 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1946	. . . . 5 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
5		cbvopab.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
6	4, 5	nfan 1931	. . . 4 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7		nfv 1946	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑧, 𝑤⟩
8		cbvopab.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfan 1931	. . . 4 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10		nfv 1946	. . . . 5 ⊢ Ⅎ𝑦 𝑣 = ⟨𝑧, 𝑤⟩
11		cbvopab.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	10, 11	nfan 1931	. . . 4 ⊢ Ⅎ𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13		opeq12 4841	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
14	13	eqeq2d 2771	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩))
15		cbvopab.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
16	14, 15	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
17	3, 6, 9, 12, 16	cbvex2v 2373	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
18	17	abbii 2827	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19		df-opab 5167	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20		df-opab 5167	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
21	18, 19, 20	3eqtr4i 2793	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811  Ⅎwnf 1815  {cab 2738  ⟨cop 4834  {copab 5166

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

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 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167

This theorem is used by:  dfrel4  6185  bj-opabco  38218  aomclem8  44172