Theorem cbvoprab1 5968

Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab1.1	⊢ Ⅎ𝑤𝜑
cbvoprab1.2	⊢ Ⅎ𝑥𝜓
cbvoprab1.3	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab1	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem cbvoprab1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab1.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1576	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 1648	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑤, 𝑦⟩
6		cbvoprab1.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	5, 6	nfan 1576	. . . . 5 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
8	7	nfex 1648	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9		opeq1 3793	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
10	9	eqeq2d 2201	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11		cbvoprab1.3	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
12	10, 11	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
13	12	exbidv 1836	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
14	4, 8, 13	cbvex 1767	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
15	14	opabbii 4085	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16		dfoprab2 5943	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		dfoprab2 5943	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
18	15, 16, 17	3eqtr4i 2220	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1471  ∃wex 1503  ⟨cop 3610  {copab 4078  {coprab 5897

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-oprab 5900

This theorem is referenced by: (None)