Theorem cbvoprab12 7457

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 1916	. . . . 5 ⊢ Ⅎ𝑤 𝑢 = ⟨𝑥, 𝑦⟩
2		cbvoprab12.1	. . . . 5 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1901	. . . 4 ⊢ Ⅎ𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1916	. . . . 5 ⊢ Ⅎ𝑣 𝑢 = ⟨𝑥, 𝑦⟩
5		cbvoprab12.2	. . . . 5 ⊢ Ⅎ𝑣𝜑
6	4, 5	nfan 1901	. . . 4 ⊢ Ⅎ𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑢 = ⟨𝑤, 𝑣⟩
8		cbvoprab12.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦 𝑢 = ⟨𝑤, 𝑣⟩
11		cbvoprab12.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
12	10, 11	nfan 1901	. . . 4 ⊢ Ⅎ𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13		opeq12 4833	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
14	13	eqeq2d 2748	. . . . 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 633	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
17	3, 6, 9, 12, 16	cbvex2v 2349	. . 3 ⊢ (∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
18	17	opabbii 5167	. 2 ⊢ {⟨𝑢, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19		dfoprab2 7426	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20		dfoprab2 7426	. 2 ⊢ {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
21	18, 19, 20	3eqtr4i 2770	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  Ⅎwnf 1785  ⟨cop 4588  {copab 5162  {coprab 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-oprab 7372

This theorem is referenced by:  cbvmpox  7461  dfoprab4f  8010  fmpox  8021  tposoprab  8214  f1od2  32813  cbvmpox2  48700