Theorem cbvoprab3 7472

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 1924	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab3.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1909	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 2346	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5	4	nfex 2346	. . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6		nfv 1924	. . . . . 6 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩
7		cbvoprab3.2	. . . . . 6 ⊢ Ⅎ𝑧𝜓
8	6, 7	nfan 1909	. . . . 5 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
9	8	nfex 2346	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
10	9	nfex 2346	. . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11		cbvoprab3.3	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))
12	11	anbi2d 638	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13	12	2exbidv 1934	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
14	5, 10, 13	cbvopab2 5166	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15		dfoprab2 7439	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16		dfoprab2 7439	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
17	14, 15, 16	3eqtr4i 2785	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550  ∃wex 1789  Ⅎwnf 1793  ⟨cop 4578  {copab 5152  {coprab 7382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-opab 5153  df-oprab 7385

This theorem is referenced by:  tposoprab  8226  erovlem  8779