Theorem cbvoprab1 6040

Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab1.1	⊢ Ⅎ𝑤𝜑
cbvoprab1.2	⊢ Ⅎ𝑥𝜓
cbvoprab1.3	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab1	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem cbvoprab1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1552	. . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩
2		cbvoprab1.1	. . . . . 6 ⊢ Ⅎ𝑤𝜑
3	1, 2	nfan 1589	. . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4	3	nfex 1661	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5		nfv 1552	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑤, 𝑦⟩
6		cbvoprab1.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	5, 6	nfan 1589	. . . . 5 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
8	7	nfex 1661	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9		opeq1 3833	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
10	9	eqeq2d 2219	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11		cbvoprab1.3	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
12	10, 11	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
13	12	exbidv 1849	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
14	4, 8, 13	cbvex 1780	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
15	14	opabbii 4127	. 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16		dfoprab2 6015	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17		dfoprab2 6015	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
18	15, 16, 17	3eqtr4i 2238	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1484  ∃wex 1516  ⟨cop 3646  {copab 4120  {coprab 5968

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-oprab 5971

This theorem is referenced by: (None)