Theorem cbvoprab13vw 36199

Description: Change the first and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvoprab13vw.1	⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab13vw	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒}

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

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

Proof of Theorem cbvoprab13vw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4897	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
4	2, 3	opeq12d 4905	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩)
5	4	eqeq2d 2751	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩))
6		cbvoprab13vw.1	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
7	5, 6	anbi12d 631	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2038	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
9	8	exbidv 1920	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
10	9	cbvexvw 2036	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒))
11	10	abbii 2812	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)}
12		df-oprab 7447	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
13		df-oprab 7447	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)}
14	11, 12, 13	3eqtr4i 2778	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 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)