Theorem cbvopabdavw 36482

Description: Change bound variables in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem cbvopabdavw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
2		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
3	1, 2	opeq12d 4839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
4	3	eqeq2d 2748	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑡 = ⟨𝑥, 𝑦⟩ ↔ 𝑡 = ⟨𝑧, 𝑤⟩))
5		cbvopabdavw.1	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))
6	4, 5	anbi12d 633	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
7	6	cbvexdvaw 2041	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2041	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
9	8	abbidv 2803	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)})
10		df-opab 5163	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
11		df-opab 5163	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2797	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑤⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ⟨cop 4588  {copab 5162

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-opab 5163

This theorem is referenced by: (None)