Theorem cbvoprab123davw 36232

Description: Change all bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvoprab123davw.1	⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab123davw	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒})

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤,𝑢,𝑣   𝜓,𝑤,𝑢,𝑣   𝜒,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑤,𝑣,𝑢)

Proof of Theorem cbvoprab123davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) → 𝑥 = 𝑤)
2		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) → 𝑦 = 𝑢)
3	1, 2	opeq12d 4905	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑢⟩)
4	3	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑢⟩)
5		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
6	4, 5	opeq12d 4905	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩)
7	6	eqeq2d 2751	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩))
8		cbvoprab123davw.1	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 631	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)))
10	9	cbvexdvaw 2038	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑣(𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)))
11	10	cbvexdvaw 2038	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑢∃𝑣(𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)))
12	11	cbvexdvaw 2038	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑢∃𝑣(𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)))
13	12	abbidv 2811	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑢∃𝑣(𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)})
14		df-oprab 7447	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
15		df-oprab 7447	. 2 ⊢ {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑢∃𝑣(𝑡 = ⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∧ 𝜒)}
16	13, 14, 15	3eqtr4g 2805	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777  {cab 2717  ⟨cop 4654  {coprab 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-oprab 7447

This theorem is referenced by: (None)