Theorem cbvoprab3 5600

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvoprab3.1	⊢ Ⅎ𝑤𝜑
cbvoprab3.2	⊢ Ⅎ𝑧𝜓
cbvoprab3.3	⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab3	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

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

Proof of Theorem cbvoprab3

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab3.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1497	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 1568	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 1568	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
7		cbvoprab3.2	. . . . . 6 ⊢ Ⅎ𝑧𝜓
8	6, 7	nfan 1497	. . . . 5 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
9	8	nfex 1568	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
10	9	nfex 1568	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11		cbvoprab3.3	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))
12	11	anbi2d 451	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13	12	2exbidv 1789	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
14	5, 10, 13	cbvopab2 3852	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15		dfoprab2 5572	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16		dfoprab2 5572	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
17	14, 15, 16	3eqtr4i 2111	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnf 1389  ∃wex 1421  ⟨cop 3401  {copab 3838  {coprab 5533

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-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

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-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536

This theorem is referenced by:  cbvoprab3v  5601  tposoprab  5918  erovlem  6221