Theorem cbvoprab2vw 36807

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 4841	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
2	1	opeq1d 4846	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
3	2	eqeq2d 2776	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
4		cbvoprab2vw.1	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
6	5	exbidv 1954	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
7	6	cbvexvw 2070	. . . 4 ⊢ (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒))
8	7	exbii 1881	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒))
9	8	abbii 2832	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)}
10		df-oprab 7423	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 7423	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4i 2798	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2743  ⟨cop 4597  {coprab 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-oprab 7423

This theorem is used by:  cbvmpo2vw2  36813