Theorem cbvopab 4104

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 1542	. . . . 5 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvopab.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1579	. . . 4 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1542	. . . . 5 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
5		cbvopab.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
6	4, 5	nfan 1579	. . . 4 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7		nfv 1542	. . . . 5 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑧, 𝑤⟩
8		cbvopab.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfan 1579	. . . 4 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10		nfv 1542	. . . . 5 ⊢ Ⅎ𝑦 𝑣 = ⟨𝑧, 𝑤⟩
11		cbvopab.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	10, 11	nfan 1579	. . . 4 ⊢ Ⅎ𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13		opeq12 3810	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
14	13	eqeq2d 2208	. . . . 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 473	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
17	3, 6, 9, 12, 16	cbvex2 1937	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
18	17	abbii 2312	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19		df-opab 4095	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20		df-opab 4095	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
21	18, 19, 20	3eqtr4i 2227	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1474  ∃wex 1506  {cab 2182  ⟨cop 3625  {copab 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095

This theorem is referenced by:  cbvopabv  4105  opelopabsb  4294