Theorem cbvopabdavw 36494

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 774	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
2		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
3	1, 2	opeq12d 4812	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
4	3	eqeq2d 2750	. . . . . 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 638	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
7	6	cbvexdvaw 2046	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2046	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
9	8	abbidv 2805	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)})
10		df-opab 5135	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
11		df-opab 5135	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2799	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑤⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786  {cab 2717  ⟨cop 4561  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5135

This theorem is referenced by: (None)