Theorem cbvopabdavw 36837

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 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
2		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
3	1, 2	opeq12d 4848	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
4	3	eqeq2d 2776	. . . . . 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 644	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
7	6	cbvexdvaw 2072	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2072	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)))
9	8	abbidv 2831	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)})
10		df-opab 5176	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
11		df-opab 5176	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑧∃𝑤(𝑡 = ⟨𝑧, 𝑤⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2825	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑤⟩ ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2743  ⟨cop 4597  {copab 5175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176

This theorem is used by: (None)