Theorem cbvoprab23davw 37189

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

Hypothesis

Ref	Expression
cbvoprab23davw.1	⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab23davw	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒})

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

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

Proof of Theorem cbvoprab23davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2761	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑥 = 𝑥)
2		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑦 = 𝑤)
3	1, 2	opeq12d 4847	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
4		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
5	3, 4	opeq12d 4847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩)
6	5	eqeq2d 2771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩))
7		cbvoprab23davw.1	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
8	6, 7	anbi12d 644	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
9	8	cbvexdvaw 2071	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
10	9	cbvexdvaw 2071	. . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
11	10	exbidv 1953	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
12	11	abbidv 2826	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)})
13		df-oprab 7428	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
14		df-oprab 7428	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
15	12, 13, 14	3eqtr4g 2820	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811  {cab 2738  ⟨cop 4834  {coprab 7425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-oprab 7428

This theorem is used by: (None)