Theorem cbvoprab2 6104

Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab2.1	⊢ Ⅎ𝑤𝜑
cbvoprab2.2	⊢ Ⅎ𝑦𝜓
cbvoprab2.3	⊢ (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

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

Proof of Theorem cbvoprab2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1577	. . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
2		cbvoprab2.1	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1614	. . . . . 6 ⊢ Ⅎ𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
4	3	nfex 1686	. . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5		nfv 1577	. . . . . . 7 ⊢ Ⅎ𝑦 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩
6		cbvoprab2.2	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
7	5, 6	nfan 1614	. . . . . 6 ⊢ Ⅎ𝑦(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
8	7	nfex 1686	. . . . 5 ⊢ Ⅎ𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
9		opeq2 3868	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
10	9	opeq1d 3873	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
11	10	eqeq2d 2243	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
12		cbvoprab2.3	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 473	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
14	13	exbidv 1873	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
15	4, 8, 14	cbvex 1804	. . . 4 ⊢ (∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
16	15	exbii 1654	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
17	16	abbii 2347	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
18		df-oprab 6032	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19		df-oprab 6032	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
20	17, 18, 19	3eqtr4i 2262	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  Ⅎwnf 1509  ∃wex 1541  {cab 2217  ⟨cop 3676  {coprab 6029

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-oprab 6032

This theorem is referenced by: (None)