Theorem cbvoprab3 7344

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 1918	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab3.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1903	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 2322	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 2322	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
7		cbvoprab3.2	. . . . . 6 ⊢ Ⅎ𝑧𝜓
8	6, 7	nfan 1903	. . . . 5 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
9	8	nfex 2322	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
10	9	nfex 2322	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11		cbvoprab3.3	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))
12	11	anbi2d 628	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13	12	2exbidv 1928	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
14	5, 10, 13	cbvopab2 5147	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15		dfoprab2 7311	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16		dfoprab2 7311	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
17	14, 15, 16	3eqtr4i 2776	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783  Ⅎwnf 1787  ⟨cop 4564  {copab 5132  {coprab 7256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-oprab 7259

This theorem is referenced by:  cbvoprab3v  7345  tposoprab  8049  erovlem  8560