Theorem cbvoprab1davw 36487

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 4831	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
2	1	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
3	2	opeq1d 4837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩)
4	3	eqeq2d 2748	. . . . . 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 633	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
7	6	2exbidv 1926	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2041	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
9	8	abbidv 2803	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)})
10		df-oprab 7372	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 7372	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑧(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2797	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ⟨cop 4588  {coprab 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7372

This theorem is referenced by:  cbvmpo1davw2  36508