Theorem cbvoprab1davw 36313

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

Hypothesis

Ref	Expression
cbvoprab1davw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab1davw	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

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

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

Proof of Theorem cbvoprab1davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4822	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
2	1	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
3	2	opeq1d 4828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩)
4	3	eqeq2d 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩))
5		cbvoprab1davw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜒))
6	4, 5	anbi12d 632	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
7	6	2exbidv 1925	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2040	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
9	8	abbidv 2797	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)})
10		df-oprab 7350	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 7350	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2791	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780  {cab 2709  ⟨cop 4579  {coprab 7347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-oprab 7350

This theorem is referenced by:  cbvmpo1davw2  36334