Theorem cbvoprab1 7506

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 1947	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab1.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1932	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 2359	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑤, 𝑦⟩
6		cbvoprab1.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	5, 6	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
8	7	nfex 2359	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9		opeq1 4840	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
10	9	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11		cbvoprab1.3	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
12	10, 11	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
13	12	exbidv 1954	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
14	4, 8, 13	cbvexv1 2376	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
15	14	opabbii 5180	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16		dfoprab2 7477	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		dfoprab2 7477	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
18	15, 16, 17	3eqtr4i 2798	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  Ⅎwnf 1816  ⟨cop 4597  {copab 5175  {coprab 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-oprab 7423

This theorem is used by:  cbvmpo1  45876