Theorem cbvoprab2 7497

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 1918	. . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
2		cbvoprab2.1	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
4	3	nfex 2318	. . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑦 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩
6		cbvoprab2.2	. . . . . . 7 ⊢ Ⅎ𝑦𝜓
7	5, 6	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑦(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
8	7	nfex 2318	. . . . 5 ⊢ Ⅎ𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
9		opeq2 4875	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
10	9	opeq1d 4880	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
11	10	eqeq2d 2744	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
12		cbvoprab2.3	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 632	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
14	13	exbidv 1925	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
15	4, 8, 14	cbvexv1 2339	. . . 4 ⊢ (∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
16	15	exbii 1851	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
17	16	abbii 2803	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
18		df-oprab 7413	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19		df-oprab 7413	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑥∃𝑤∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
20	17, 18, 19	3eqtr4i 2771	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782  Ⅎwnf 1786  {cab 2710  ⟨cop 4635  {coprab 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-oprab 7413

This theorem is referenced by:  cbvmpo2  43786