Theorem cbvoprab13vw 37157

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

Hypothesis

Ref	Expression
cbvoprab13vw.1	⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab13vw	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒}

Distinct variable groups:   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 cbvoprab13vw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
2	1	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
3		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
4	2, 3	opeq12d 4850	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩)
5	4	eqeq2d 2774	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩))
6		cbvoprab13vw.1	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
7	5, 6	anbi12d 644	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2073	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
9	8	exbidv 1955	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)))
10	9	cbvexvw 2071	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒))
11	10	abbii 2830	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)}
12		df-oprab 7431	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
13		df-oprab 7431	. 2 ⊢ {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑤∃𝑦∃𝑣(𝑡 = ⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∧ 𝜒)}
14	11, 12, 13	3eqtr4i 2796	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811  {cab 2741  ⟨cop 4837  {coprab 7428

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

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 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-oprab 7431

This theorem is used by: (None)