Theorem cbvoprab12 5945

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
cbvoprab12.1	⊢ Ⅎ𝑤𝜑
cbvoprab12.2	⊢ Ⅎ𝑣𝜑
cbvoprab12.3	⊢ Ⅎ𝑥𝜓
cbvoprab12.4	⊢ Ⅎ𝑦𝜓
cbvoprab12.5	⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab12	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣

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

Proof of Theorem cbvoprab12

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1528	. . . . 5 ⊢ Ⅎ𝑤 𝑢 = ⟨𝑥, 𝑦⟩
2		cbvoprab12.1	. . . . 5 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1565	. . . 4 ⊢ Ⅎ𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1528	. . . . 5 ⊢ Ⅎ𝑣 𝑢 = ⟨𝑥, 𝑦⟩
5		cbvoprab12.2	. . . . 5 ⊢ Ⅎ𝑣𝜑
6	4, 5	nfan 1565	. . . 4 ⊢ Ⅎ𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 𝑢 = ⟨𝑤, 𝑣⟩
8		cbvoprab12.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfan 1565	. . . 4 ⊢ Ⅎ𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10		nfv 1528	. . . . 5 ⊢ Ⅎ𝑦 𝑢 = ⟨𝑤, 𝑣⟩
11		cbvoprab12.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	10, 11	nfan 1565	. . . 4 ⊢ Ⅎ𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13		opeq12 3780	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
14	13	eqeq2d 2189	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝑢 = ⟨𝑥, 𝑦⟩ ↔ 𝑢 = ⟨𝑤, 𝑣⟩))
15		cbvoprab12.5	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))
16	14, 15	anbi12d 473	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
17	3, 6, 9, 12, 16	cbvex2 1922	. . 3 ⊢ (∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
18	17	opabbii 4069	. 2 ⊢ {⟨𝑢, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19		dfoprab2 5918	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20		dfoprab2 5918	. 2 ⊢ {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
21	18, 19, 20	3eqtr4i 2208	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  Ⅎwnf 1460  ∃wex 1492  ⟨cop 3595  {copab 4062  {coprab 5872

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4064  df-oprab 5875

This theorem is referenced by:  cbvoprab12v  5946  cbvmpox  5949  dfoprab4f  6190  fmpox  6197  tposoprab  6277