Theorem cbvoprab2vw 36466

Description: Change the second bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvoprab2vw.1	⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab2vw	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒}

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

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

Proof of Theorem cbvoprab2vw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4805	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
2	1	opeq1d 4810	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
3	2	eqeq2d 2750	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
4		cbvoprab2vw.1	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 638	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
6	5	exbidv 1928	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
7	6	cbvexvw 2044	. . . 4 ⊢ (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒))
8	7	exbii 1855	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒))
9	8	abbii 2806	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)}
10		df-oprab 7360	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 7360	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4i 2772	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒}

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786  {cab 2717  ⟨cop 4561  {coprab 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-oprab 7360

This theorem is referenced by:  cbvmpo2vw2  36472