Theorem cbvopab2 3852

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvopab2.1	⊢ Ⅎ𝑧𝜑
cbvopab2.2	⊢ Ⅎ𝑦𝜓
cbvopab2.3	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopab2	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧

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

Proof of Theorem cbvopab2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩
2		cbvopab2.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1497	. . . . 5 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩
5		cbvopab2.2	. . . . . 6 ⊢ Ⅎ𝑦𝜓
6	4, 5	nfan 1497	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)
7		opeq2 3571	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
8	7	eqeq2d 2092	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑧⟩))
9		cbvopab2.3	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
10	8, 9	anbi12d 456	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)))
11	3, 6, 10	cbvex 1679	. . . 4 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
12	11	exbii 1536	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
13	12	abbii 2194	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
14		df-opab 3840	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15		df-opab 3840	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
16	13, 14, 15	3eqtr4i 2111	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnf 1389  ∃wex 1421  {cab 2067  ⟨cop 3401  {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840

This theorem is referenced by:  cbvoprab3  5600