Theorem cbvoprab13davw 36520

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

Hypothesis

Ref	Expression
cbvoprab13davw.1	⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab13davw	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒})

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

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

Proof of Theorem cbvoprab13davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑥 = 𝑤)
2		eqidd 2742	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑦 = 𝑦)
3	1, 2	opeq12d 4815	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
4		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
5	3, 4	opeq12d 4815	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩)
6	5	eqeq2d 2752	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩))
7		cbvoprab13davw.1	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
8	6, 7	anbi12d 639	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
9	8	cbvexdvaw 2047	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
10	9	exbidv 1929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑤) → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
11	10	cbvexdvaw 2047	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
12	11	abbidv 2807	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)})
13		df-oprab 7364	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
14		df-oprab 7364	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)}
15	12, 13, 14	3eqtr4g 2801	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787  {cab 2719  ⟨cop 4564  {coprab 7361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-oprab 7364

This theorem is referenced by: (None)