Theorem cbvopab 3931

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 1473	. . . . 5 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvopab.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1509	. . . 4 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1473	. . . . 5 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
5		cbvopab.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
6	4, 5	nfan 1509	. . . 4 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7		nfv 1473	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑧, 𝑤⟩
8		cbvopab.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfan 1509	. . . 4 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10		nfv 1473	. . . . 5 ⊢ Ⅎ𝑦 𝑣 = ⟨𝑧, 𝑤⟩
11		cbvopab.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	10, 11	nfan 1509	. . . 4 ⊢ Ⅎ𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13		opeq12 3646	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
14	13	eqeq2d 2106	. . . . 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 458	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
17	3, 6, 9, 12, 16	cbvex2 1852	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
18	17	abbii 2210	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19		df-opab 3922	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20		df-opab 3922	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
21	18, 19, 20	3eqtr4i 2125	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  Ⅎwnf 1401  ∃wex 1433  {cab 2081  ⟨cop 3469  {copab 3920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-opab 3922

This theorem is referenced by:  cbvopabv  3932  opelopabsb  4111