Theorem cbvoprab1 7502

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 1913	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab1.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1898	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 2323	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑤, 𝑦⟩
6		cbvoprab1.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	5, 6	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
8	7	nfex 2323	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9		opeq1 4853	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
10	9	eqeq2d 2745	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11		cbvoprab1.3	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
12	10, 11	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
13	12	exbidv 1920	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
14	4, 8, 13	cbvexv1 2342	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
15	14	opabbii 5190	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16		dfoprab2 7473	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		dfoprab2 7473	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
18	15, 16, 17	3eqtr4i 2767	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778  Ⅎwnf 1782  ⟨cop 4612  {copab 5185  {coprab 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5186  df-oprab 7417

This theorem is referenced by:  cbvmpo1  45060