Theorem cbvoprab1vw 36597

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

Hypothesis

Ref	Expression
cbvoprab1vw.1	⊢ (𝑥 = 𝑤 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab1vw	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒}

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

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

Proof of Theorem cbvoprab1vw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4831	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
2	1	opeq1d 4837	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩)
3	2	eqeq2d 2773	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩))
4		cbvoprab1vw.1	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 641	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
6	5	2exbidv 1944	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
7	6	cbvexvw 2057	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒))
8	7	abbii 2829	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
9		df-oprab 7400	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
10		df-oprab 7400	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
11	8, 9, 10	3eqtr4i 2795	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560  ∃wex 1799  {cab 2740  ⟨cop 4588  {coprab 7397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7400

This theorem is referenced by:  cbvmpo1vw2  36603