Theorem cbvoprab3 6021

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 1551	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab3.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1588	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 1660	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 1660	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6		nfv 1551	. . . . . 6 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
7		cbvoprab3.2	. . . . . 6 ⊢ Ⅎ𝑧𝜓
8	6, 7	nfan 1588	. . . . 5 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
9	8	nfex 1660	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
10	9	nfex 1660	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11		cbvoprab3.3	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))
12	11	anbi2d 464	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13	12	2exbidv 1891	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
14	5, 10, 13	cbvopab2 4118	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15		dfoprab2 5992	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16		dfoprab2 5992	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
17	14, 15, 16	3eqtr4i 2236	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1483  ∃wex 1515  ⟨cop 3636  {copab 4104  {coprab 5945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948

This theorem is referenced by:  cbvoprab3v  6022  tposoprab  6366  erovlem  6714